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This study aims to explore in-service preschool teachers' attitudes towards and beliefs about geometry and its teaching and to investigate their knowledge of shapes and solids. The study uses a mixed descriptive approach employing quantitative and qualitative research methods. Thirty-four Israeli preschool teachers of children aged 3-6 from 34 different preschools and kindergartens participated in the study and responded to a questionnaire comprised of closed and open items. Regarding their attitudes and beliefs, the preschool teachers were asked about: the meaning of geometry; affinity for geometry, importance of geometry; need to use accurate mathematical language, and about their enjoyment of engaging young children in activities connected with learning geometry. As to the knowledge of shapes and solids, the preschool teachers were asked to describe or define shapes and solids, name shapes and solids and distinguish between rectangles and other shapes. The results illustrate that most of the preschool teachers like geometry or expressed a neutral position towards geometry, but some hate geometry. Most of the preschool teachers comprehend the importance of engaging in activities that relate to geometry in preschool and enjoy involving young children in activities connected with learning geometry. As to the use of accurate language, most of the preschool teachers maintain that accurate mathematical language should be applied when preschool children aged 3-6 deal with shapes and solids. Yet, only about half of them believe that it is necessary to use accurate language when the children are at an earlier age. Certain preschool teachers lack the knowledge to name shapes and solids, even those which are part the mathematics curriculum for ages 3-6. They exhibited difficulties in using accurate mathematical language when describing shapes and solids, and were influenced by the shapes' visual appearance rather than by their definitions and properties. It is recommended that preschool teachers attend further development programs so they can appropriately involve young children in activities connected to learning geometry, as required by the curriculum.

Spatial ability was defined by Lohman (1996) as the ability to create, maintain, retrieve, and change visual pictures. Linn and Peterson (1985) distinguished among spatial perception, mental rotation, and spatial visualization. Spatial perception is the determination of spatial connections with respect to the orientation of the learners' body; mental rotation is the ability to accurately and rapidly rotate mentally two- or three-dimensional (2D or 3D) geometric figures, and spatial visualization is related to the ability to identify multistep complex manipulations of spatial information. Among these three types of spatial skills, the ability to perform a mental rotation has the closest relation to success in the science, technology, engineering, and mathematics (STEM) fields. In many education systems around the globe, solid geometry constitutes an integral part of the mathematics curriculum for different ages and at different learning levels. Nevertheless, the topic of mental rotation is not always explicitly manifested in the curriculum. Given the importance attributed to the ability to perform a mental rotation, the authors present several activities intended to develop this ability among learners in grades 6-8. The activities constitute an example of tasks that can be continuously integrated within the routine teaching framework of mathematics lessons. Alternately, these tasks can be assigned as homework, to allow learners a longer time for developing strategies that will help them to deal with the embedded challenges.

This study explores effects of teaching experience and specialized mathematics education on mathematics anxiety and mathematics teaching anxiety. Fifty-nine female pre-service and in-service mathematics primary school teachers with varied specialized mathematics education and teaching experience responded to a questionnaire measuring mathematics anxiety and mathematics teaching anxiety. The results illustrate that specialized mathematics education affects in-service primary school teachers' mathematics anxiety, and that teaching experience is associated with mathematics teaching anxiety. Moreover, mathematics anxiety of some in-service primary teachers without specialization does not fade with time. The discussion highlights the importance of identifying mathematics anxiety in primary school teachers with no specialized mathematics education, aiming to lower it. The small sample and participants' background limit the power of the findings. This study recommends conducting further studies according to the suggested mathematics anxiety cognitive-behavioral model; addressing cultural differences, teachers with and without specialized mathematics education, and comparing middle and secondary teachers.

Despite the plethora of research in the field of mental rotation, researchers are still divided over a range of related issues, particularly the processes involved in coping with mental rotation tasks. This field of research, for the most part, is descriptive rather than interpretative, leading to the absence of a theoretical framework to enable the development of a coherent dedicated training program. Against this background, the present study is underpinned by the assumption that identifying relations between strategies that potentially lead to a successful coping with mental rotation tasks and those that do not lead to success might help us in designing a dedicated training program for pre-service teachers. The study involved 122 pre-service elementary school mathematics teachers. They responded to a questionnaire that included, among others, two tasks that required performing mental rotation. In line with other studies, the results of the present study indicate that there is an advantage to implementing a holistic strategy over an analytic one. However, the mere implementation of a certain strategy does not guarantee that one would draw a correct conclusion, as the same strategy might lead to either correct or incorrect conclusion. Based on these results, we inferred that there is a mechanism that mediates between the strategy employed and the conclusion reached. Apparently, in the context of mental rotation, the study of strategies does not allow understanding the essence of this mechanism, and in order to construct an interpretative theory, an alternative course of inquiry is needed.

Acquaintance with various ways of inculcating concepts in any studied area of knowledge is one of teachers' duties, particularly mathematics teachers. Studies indicate errors and difficulties when inculcating concepts in mathematics and learning them. Many concepts have different meanings in different contexts. Hence, teachers should deal with the image of the concept by paying attention to its essence. They should also deal with misconceptions developed while inculcating the concept. The present paper describes an activity conducted with 28 teachers studying for an M.Ed. degree in mathematics education at a teacher education college. They attend a course of solid geometry, a topic which is perceived as extremely difficult for the learners. The activity focuses on two ways of inculcating a solid geometry concept: "an angle between a lateral face and a base of a pyramid" as one of the ways for developing spatial orientation. Upon completion of the activity, the paper illustrates the participants' responses regarding their perception of the advantages of each of the ways they experienced and their recommendations as to their preferred ways for inculcating new geometric concepts.

In this study, we aim to explore the extent of mathematics pre-service teachers' ability to apply their procedural understanding combined with spatial perception for drawing conceptual conclusions related to angles in a pyramid. The participants are 16 pre-service high school mathematics teachers. They have studied solid geometry during one academic year, solving problems with various 3-D geometric figures including pyramids and engaging in activities designed to develop spatial perception. At the end of the year, they have taken a final test which examines procedural understanding of 3-D geometric figures as well as relational understanding and spatial perception regarding angles in pyramids. The results illustrate that attainments of the majority of the pre-service teachers in problems requiring only procedural understanding are higher than the attainments in problems which require relational understanding. The results also lead to the assumption that relational understanding of learned material requires application of special teaching methods. Hence, we recommend integrating in the syllabus appropriate courses that focus on the development of this type of understanding.

From a young age children feel the need to identify two-dimensional geometric figures (shapes) and three-dimensional geometric figures (solids). The National Council of Teachers of Mathematics indicates the importance of being able to identify and name various geometric figures by kindergarten age. One of the objectives of this study was to learn the ability of second graders to identify examples and non-examples of three generally known solids: cylinder, cone and pyramid, and to justify their identification based on the attributes (critical and non-critical) of those solids. Another objective was to find out whether changing the position of the solids would result in those children maintaining their decisions regarding the name or changing their identification of the solids, giving arguments accordingly. Findings of this study illustrate that children can identify and characterise solids presented to them in a typical position. However, they find it difficult to correctly identify the same solids in another position. An interesting finding was that most of the arguments given to justify their identification were based on the specific attributes of the solid rather than on the perception of the solids in general. Findings of the present study suggest that it is highly important for learners to be acquainted with a variety of both non-examples and examples of solids. Moreover, it is recommended that solids are presented to learners not only in the typical position, in order to improve their ability to identify them and understand that the name of the figure does not change when its position changes.

This paper presents the major findings and recommendations from a formative evaluation of the first of a 2-year in-service training program. This program was designed for the professional development of math teachers. It was developed following an educational reform, which was meant to enhance the teachers' ability to generate changes in their teaching methods and implement them in class. The teachers' attitudes towards the program were examined using close-ended questionnaires and open-ended questions which were administered to 19 teachers at the end of their first year of inclusion in the program. The research findings attested to the contribution of the program to the teachers' development - mainly to those with longer experience in math teaching -- in the area of the taught discipline itself, as well as in the didactic-educational area and the personal-emotional area. These findings were anchored in a conceptual framework of a model of the development of the teachers' nature of concern and interest regarding the change. The article discusses the potential instrumental and conceptual application of the findings in a focused decision-making process pertaining to running the program, and to a research-based accompaniment of the implementation process of professional development programs.

This study explored Israeli elementary school mathematics teachers' perceptions of their professional needs, with the purpose of developing in-service training courses which cater to these needs. Eighty-four teachers responded to the questionnaire and were interviewed. The results indicate that the respondents' main needs are associated with strengthening their didactical knowledge capability of dealing with emotional aspects that relate to pupils' learning of mathematics. Although most of the respondents lack formal mathematics education, they attribute less importance to their need to enhance knowledge in the field. In light of Israeli pupils' relatively low attainments in mathematics, the authors believe that the education system should require teachers to expand their mathematics knowledge and that only teachers with appropriate knowledge will be permitted to teach the discipline.

The present study aimed to explore the contribution of a mathematics in-service training course to elementary school teachers (1st-6th grades) in Israel. The study was conducted among 449 educators. They were required to respond to background questions. Moreover, they were asked to indicate their expectations from the in-service training course and, at its end, point out to what extent they benefitted from that course. The research findings illustrate that educators who teach mathematics at elementary school and attended the course are generally women in their 40s, holding a BEd degree and a teaching certificate "not" in mathematics, with an average 13-year seniority. The participating teachers indicated their wish to enrich their didactic knowledge in order to acquire varied tools for teaching mathematics to the entire pupil population as well as to gifted pupils and pupils with learning difficulties. Nevertheless, their demand to expand their mathematics knowledge was very limited. Based on the fact that most teachers have no mathematics education, this is a surprising finding as, in order to be a good teacher, one must be versed not only in Pedagogical Content Knowledge but also in Subject Matter Knowledge.

The present paper engages in a qualitative research of self-assessment of two lecturers and their students within the framework of a mathematics teaching seminar course (a course during which students submit a research final work) at a teachers' training college in Israel. Two lecturers co-teach in the course - one of them in the discipline of mathematics and the other in the discipline of academic literacy. The study explores the development process of this experience in the course of one academic year. Data were gathered by means of observations, interviews and reflection records, and the data analysis was done by the "grounded theory" method. The research findings indicate that the self-assessment experience of students is far from being faultless. (Contains 2 tables.)

This case study examines the effect of coteaching on the development of a student's professional knowledge. The study analyzes the student's perceptions of her experiences in a yearlong seminar course of math teaching in elementary school, conducted at a teachers college in Israel. The course was interdisciplinary and taught jointly by two lecturers--one from the discipline of mathematics and the other from the discipline of academic literacy. Analysis of observations, interviews, and documents illustrates the change in the student's perception of her professional knowledge--both academic and practical--regarding math teaching, as well as the contribution of caring coteaching to this change. The main conclusion of the study is that caring coteaching considerably enhanced the student's professional knowledge. This conclusion is in line with the rather limited literature dealing with this topic. (Contains 1 figure and 1 note.)

This case study of one class versus a control group focused on the impact of an intervention unit, which is not part of the regular curriculum, on the improvement of spatial ability of high school students (forty-six 12th-graders, aged 17-18, both boys and girls) in general as well as from a gender perspective. The study explored three sub-abilities: mental rotation (MR), spatial visualization (VS) and spatial orientation (SO). Findings indicated that the spatial orientation of the experimental group students had considerably improved. The findings also illustrated a significant gender-based advantage in favour of the boys in some of the spatial abilities even before the implementation of the intervention unit. The hypothesis relating to the reduction of the gender differences was not corroborated. (Contains 11 figures and 3 tables.)

The paper presents findings of a small scale study of a few items related to problem solving with squares and roots, for different teacher groups (pre-service and in-service mathematics teachers: elementary and junior high school). The research participants were asked to explain what would be the units digit of a natural number to be squared in order to obtain a certain units digit as a result. They were also asked to formulate a rule--an algorithm for calculating the square of a 2-digit number which is a multiple of 5. Based on this knowledge and estimation capability, they were required to find, without using calculators, the square roots of given natural numbers. The findings show that most of the participants had only partial intuition regarding the units' digit of a number which is squared when the units' digit of the square is known. At the same time, the participants manifested some evidence of creativity and flow of ideas in identifying the rule for calculating the square of a natural number whose units digit is 5. However, when asked to identify, by means of estimation and based on knowledge from previous items, the square roots of three natural numbers, only few of them managed to find the three roots by estimation.

Proving and refuting mathematical claims constitute a significant element in the development of deductive thinking. These issues are mainly studied during geometry lessons and very little (if at all) in lessons of other mathematical disciplines. This study deals with high school students' perceptions of proofs in the geometry. The study explores whether students know when to use a deductive proof and when an example is sufficient for proving or refuting geometrical claims. The findings indicate that in cases of simple claims, the students corroborate them by using a deductive proof. However, when the claim is more complex, the students tend to present both a proof and an example. Moreover, they are unsure whether using an example can constitute a method for proving a mathematical claim, believing that in mathematics everything must be proven. They believe that examples are used merely for illustration purposes rather than as a means of convincing. The research conclusions support the need for deepening and developing the students' distinction between cases where examples are insufficient and cases where an example is sufficient for proving a claim. (Contains 2 tables and 3 figures.)

The article aims to check the way adults, "some who are" practicing mathematics teachers at elementary school, "some who are" academicians making a career change to mathematics teachers at junior high school and the "rest who are" pre-service mathematics teachers at elementary school, cope with the solution of everyday real-world problems of buying and selling. The findings show that even adults with mathematical background tend to make mistakes in solving "everyday" real-world "problems". Only about 70% of the adults who have an orientation to mathematics solved the "sample problem" correctly. The lowest percentage of success was demonstrated by the academicians making a career change to junior high school mathematics teachers whereas the highest percentage of success was manifested by pre-service elementary school mathematics teachers. Moreover, the findings illustrate that life experience of the practicing mathematics teachers and, mainly, of the academicians making a career change, who were older than the pre-service teachers, did not facilitate the solution of such a real-world problem. Perhaps the reason resides in the process of mathematics teaching at school, which does not put an emphasis on the solution of everyday real-world problems. (Contains 2 tables and 1 note.)

This article aims to present the findings of a research which investigated the effect of a difference in word formulation and mathematical characteristics of story problems on their successful solution by preservice mathematics teachers (students) and practising mathematics teachers. The findings show that in the case of a problem with a difference in word formulation of one of its components, but that difference does not entail a difference in the mathematical characteristics and the solution process, the percentage of correct answers to the two versions did not vary significantly. However, when there is a difference in the word formulation of one of the problem's components, leading to difference in the mathematical characteristics and the problem solution, there is a wide gap in the percentage of correct answers between the two versions. These findings were relevant to the preservice mathematics teachers as well as to the practising mathematics teachers. Moreover, the findings illustrate that the students displayed greater originality in the solution process than did the practising teachers. Consequently, during in-service courses and teacher education, it is recommended that we need to focus also on story problems with different word formulations in order to facilitate the thinking processes leading to their solutions. (Contains 3 tables and 2 figures.)

The present study depicts the story of three elementary school mathematics teachers who come from a different socio-economic background but have similar entrance characteristics for college studies. They have chosen teaching as their first and sole alternative and as their preferred profession in life. All three were integrated into the education system upon graduation from college and, a short time later, less than three years, are playing key roles in the field of education in addition to their teaching practice in class. The study aims to identify unique characteristics of college graduate teachers who undergo an accelerated process of professional development. Their stories illustrate that the attitude towards the discipline (love of the profession, the wish to teach it), the wish to make an impact on what is transpiring in the system, personality variables and the school context in which they work as teachers are major and significant factors in their professional growth. (Contains 1 table, 1 figure and 1 note.)

This study reveals the multicultural perspectives in the curricula of two colleges of education in Israel. It examines the perceptions of teacher educators with regard to: the importance of the teacher's role in educating students for sensitivity and diversity; the needs of students from different cultural backgrounds in colleges of education; the roles of the teacher and the college; and an evaluation of how multiculturalism is dealt with in teacher education in Israel. Faculty members in both colleges perceived multiculturalism as relating to two domains: "student population" and "curriculum". The findings and conclusions confirm the existence and increasing awareness of multiculturalism in the two colleges. However, it is obvious that as yet the concept has not been thoroughly investigated, nor has it penetrated all realms of activity. Clearly, teacher educators know the meaning of multiculturalism and believe that it should play a more significant role in colleges of education. However, they seem to be uncertain as to how to go about it. (Contains 5 tables and 1 figure.)

This study presents the narratives of three teacher educators in Israeli colleges of education: an Israeli-born Jew, a Jewish immigrant, and an Israeli-born Arab. The personal and professional stories expose their voices, life experiences, and worldviews, and display the relations between these and their multicultural awareness. Through narrative interviews, each becomes a representative of his or her reference group. Content analysis revealed intercultural transitions and encounters, the dilemma of duality, and significant episodes and their impact on instruction. A critical essence, a main theme or leitmotif, was unique to each story. The complexity of Israeli society is manifested in the duality they all express and in their diverse perceptions regarding multiculturalism. They are all aware of multiculturalism and believe that they embody this understanding in their work as teacher educators, each doing it his or her way. (Contains 1 figure.)