Ana Kuzle
email: kuzle@uni-potsdam.de

The teaching of geometry in primary education: Is geometry still neglected in school mathematics?

Keywords: Geometry, teaching geometry, primary education, in-service teachers

Primary school mathematics has undergone many changes in the last twenty years. Geometry in particular was re-evaluated due to a paradigm change and was attributed greater importance in curricula worldwide. Despite a clear consensus that geometry instruction is an indispensable component of mathematics even in primary education, the status of elementary school geometry is still considered unsatisfactory in many cases. This report aimed to find out if earlier findings that geometry is neglected in school mathematics instruction can be confirmed based on new empirical data using a questionnaire. In total, 120 primary grade teachers participated in the study. The study confirmed the results from previous research, but also shed light on new emerging factors contributing to the unsatisfactory state of current geometry teaching in primary education.

Introduction

For several decades, researchers have emphasized the importance of teaching geometry [START_REF] Franke | Didaktik der Geometrie in der Grundschule[END_REF][START_REF] Sinclair | New opportunities in geometry education at the primary level[END_REF][START_REF] Vigilante | Geometry for primary children: Considerations[END_REF]. Engagement with geometric content promotes basic cognitive skills, allows the development of specific mathematical ways of thinking, and contributes greatly to understanding the world in which we live [START_REF] Bauersfeld | Drei Gründe, geometrisches Denken in der Grundschule zu fördern [Three reasons to promote geometric thinking in elementary school[END_REF][START_REF] Franke | Didaktik der Geometrie in der Grundschule[END_REF]. Furthermore, all mathematics is permeated by geometric ways of thinking, because whenever a piece of visual information is perceived, analyzed, and stored, geometric thinking is fundamental to this process. Despite the acknowledged importance of geometry, it seems to have lost its position in school mathematics developing the reputation of being the "problem child" of mathematics teaching as was reported by [START_REF] Backe-Neuwald | Bedeutsame Geometrie in der Grundschule: Aus Sicht der Lehrerinnen und Lehrer, des Faches, des Bildungsauftrages und des Kindes[END_REF] more than two decades ago.

Mathematics teaching in elementary schools has been influenced by many changes that have taken place in the last 20 years. Firstly, to counteract the worldwide trend of the reduction of geometry in the school curriculum [START_REF] Clements | Teaching and learning geometry[END_REF][START_REF] Van De Walle | Teaching student-centered mathematics[END_REF], the role of geometry concerning perspectives on the teaching of geometry for the 21st century has been reassessed (Mammana & Villani, 1998). Secondly, the teaching of geometry has been re-evaluated due to a paradigm shift in school education to a focus on process-oriented mathematical competencies, such as problem solving and argumentation. The fundamental question of whether geometry is being neglected remains open, arguably with good reason, as the status of elementary school geometry instruction is still considered unsatisfactory, especially compared to arithmetic [START_REF] Clements | Teaching and learning geometry[END_REF][START_REF] Franke | Didaktik der Geometrie in der Grundschule[END_REF]. Similar results were reported in the study by [START_REF] Backe-Neuwald | Bedeutsame Geometrie in der Grundschule: Aus Sicht der Lehrerinnen und Lehrer, des Faches, des Bildungsauftrages und des Kindes[END_REF] in which nearly 80% of primary grade teachers agreed with the statement that the teaching of geometry is neglected and mostly serves as entertainment rather than important mathematical content. Since these findings, however, the mathematics curriculum in Germany has been revised, making a reassessment of the empirical findings of [START_REF] Backe-Neuwald | Bedeutsame Geometrie in der Grundschule: Aus Sicht der Lehrerinnen und Lehrer, des Faches, des Bildungsauftrages und des Kindes[END_REF] necessary. Furthermore, only a few studies (e.g., [START_REF] Sitter | Geometrische Körper an inner-und außerschulischen Lernorten: Der Einfluss des Protokollierens auf eine sichere Begriffsbildung[END_REF][START_REF] Wiese | Kognitive Anforderungen in geometrischen Aufgaben des vierten Schuljahres [Cognitive demands in geometric tasks of the fourth school year[END_REF] are based on recent data subsequent to the paradigm shift. This paper takes into consideration a number of the recent discussions in the literature by focusing on the role of geometry instruction in elementary school mathematics. Specifically, utilizing two questionnaire items adapted from [START_REF] Backe-Neuwald | Bedeutsame Geometrie in der Grundschule: Aus Sicht der Lehrerinnen und Lehrer, des Faches, des Bildungsauftrages und des Kindes[END_REF], I investigate whether there are any changes in the previously reported situation regarding the neglect of geometry in school mathematics.

Related literature

The study by [START_REF] Backe-Neuwald | Bedeutsame Geometrie in der Grundschule: Aus Sicht der Lehrerinnen und Lehrer, des Faches, des Bildungsauftrages und des Kindes[END_REF] involving 108 in-service teachers represents a milestone in the contemporary empirical examination of geometry in primary schools and its role in the school structure, its results have been followed up in the form of a replication [START_REF] Sitter | Geometrische Körper an inner-und außerschulischen Lernorten: Der Einfluss des Protokollierens auf eine sichere Begriffsbildung[END_REF] and an extension study [START_REF] Wiese | Kognitive Anforderungen in geometrischen Aufgaben des vierten Schuljahres [Cognitive demands in geometric tasks of the fourth school year[END_REF]. The teachers' perceptions of school geometry were very ambivalent: when asked about associations with geometry instruction in elementary school, some teachers described it as "a welcome change" or "an exciting thing," whereas others considered it "secondary," "beside the point", or "not important" (Backe-Neuwald, 2000, p. 16-18.). As noted earlier, almost 80% of the inservice teachers surveyed believed that geometry was neglected in elementary school, not because of the curricular requirements of the syllabi, but due to issues concerning its actual implementation in the classroom [START_REF] Backe-Neuwald | Bedeutsame Geometrie in der Grundschule: Aus Sicht der Lehrerinnen und Lehrer, des Faches, des Bildungsauftrages und des Kindes[END_REF]. Only 15% of the teachers surveyed disagreed with the statement. In 89 cases (82.4%), the teachers stated that they have to utilize the available instructional time for more important mathematical content, namely arithmetic, which was identified as the main reason for the neglect of geometry in school mathematics. Moreover, many teachers (n = 65; 60.2%) shied away from the preparation intensity of geometry instruction [START_REF] Backe-Neuwald | Bedeutsame Geometrie in der Grundschule: Aus Sicht der Lehrerinnen und Lehrer, des Faches, des Bildungsauftrages und des Kindes[END_REF] which was identified as the second reason for geometry being neglected in school mathematics. This was confirmed in the work of [START_REF] Sitter | Geometrische Körper an inner-und außerschulischen Lernorten: Der Einfluss des Protokollierens auf eine sichere Begriffsbildung[END_REF]. Along these lines, the third reason reported was that too little support in the form of additional materials is available for the preparation of geometry lessons.

Teachers themselves play a crucial role when it comes to explaining the deficit in the teaching of geometry. Teacher training has been strongly criticized in the past. [START_REF] Bauersfeld | Drei Gründe, geometrisches Denken in der Grundschule zu fördern [Three reasons to promote geometric thinking in elementary school[END_REF] noted that elementary school teacher education is divided among different faculties into subject and pedagogical content knowledge which does not reflect school reality and, above all, limits development in the field of didactics. Such shortfalls are specifically attributed to geometry courses, which lead to subject-didactic uncertainties that either partially or completely prevent teachers from teaching geometry [START_REF] Sitter | Geometrische Körper an inner-und außerschulischen Lernorten: Der Einfluss des Protokollierens auf eine sichere Begriffsbildung[END_REF]. Further, [START_REF] Clements | Teaching and learning geometry[END_REF] reported that a large proportion of prospective mathematics teachers only reached Level 2 of the van Hiele model of geometric reasoning (i.e., descriptive reasoning). Similar phenomena were reported in the study by [START_REF] Backe-Neuwald | Bedeutsame Geometrie in der Grundschule: Aus Sicht der Lehrerinnen und Lehrer, des Faches, des Bildungsauftrages und des Kindes[END_REF]. The teachers viewed their own previous education with regard to geometry topics and content as insufficient (n = 28; 25.9%) and they felt insecure teaching geometry (n = 21; 19.4%). The latter was most prominent in teachers with little teaching experience as well as out-of-field teachers (i.e., nonspecialists). According to [START_REF] Backe-Neuwald | Bedeutsame Geometrie in der Grundschule: Aus Sicht der Lehrerinnen und Lehrer, des Faches, des Bildungsauftrages und des Kindes[END_REF], this leads to a high degree of professional uncertainty in the conveyance of geometric content and, ultimately, to the deprioritization of geometry lessons. Textbooks were identified as the sixth main factor (n = 18; 16.7%) contributing to the neglect of geometry because textbook suggestions were perceived as being too trivial, easy and not challenging enough. Other reasons, such as time intensive methodological preparation, intensity of subject preparation, and difficulty of student assessment, were mentioned by less than 10% of teachers. Difficulty of student assessment was also reported by [START_REF] Sitter | Geometrische Körper an inner-und außerschulischen Lernorten: Der Einfluss des Protokollierens auf eine sichere Begriffsbildung[END_REF].

Research process

For this study, a mixed-methods research design was chosen. Elementary schools were selected through existing contact with the resercher's university and random inquiries. The sample of 120 in-service primary teachers (Grades 1-6) consisted of 22 male teachers (18.3%) and 97 female teachers (80.8%). One participant did not provide gender information. A total of 90 teachers taught mathematics as subject specialists (75%) and 29 of them were non-subject specialists (24.2%). The data of one participant was not provided. In terms of professional experience, 16 teachers (13.3%) have been teaching mathematics for less than or up to two years, 18 (15%) for up to 5 years, 19 (15.8%) for up to 10 years, 16 (13.3%) for up to 20 years, and 51 (42.5%) for more than 20 years.

The main source of data was a questionnaire on the state of the teaching of geometry in primary grades that was based on an adaptation of the instrument of [START_REF] Backe-Neuwald | Bedeutsame Geometrie in der Grundschule: Aus Sicht der Lehrerinnen und Lehrer, des Faches, des Bildungsauftrages und des Kindes[END_REF]. Additionally, new items were developed based on literature published in the last 20 years (e.g., [START_REF] Clements | Teaching and learning geometry[END_REF][START_REF] Franke | Didaktik der Geometrie in der Grundschule[END_REF]. To cover a broad field of research, six sections were included. Each section consisted of several items with both open and closed questions. In addition to personal information, there were questions about the characteristics of geometry lessons, the use of materials, the goals and aspects of teaching geometry, possible reasons for the neglect of geometry, and the participants' personal attitude toward teaching geometry. The questionnaires were either distributed on site or were sent by e-mail. All questionnaires were returned to the author by mail.

In this report, I focus on the section of the questionnaire dealing with the possible neglect of geometry, which was measured with two items. Both items were analyzed in order to answer the research question. The first item was as follows: "It is frequently claimed that geometry instruction is neglected in elementary education and only leads a 'Cinderella existence' there. What is your opinion?" [START_REF] Backe-Neuwald | Bedeutsame Geometrie in der Grundschule: Aus Sicht der Lehrerinnen und Lehrer, des Faches, des Bildungsauftrages und des Kindes[END_REF]. Three options were given for the item where one response was to be marked and then justified in a free-form text response. In addition to "I agree because..." and "I do not agree because…", "Other" was provided as a third option. It was assumed that there may be teachers who are not in a position to or are unwilling to take a position. The analysis of the item was based on qualitative content analysis according to [START_REF] Mayring | Qualitative Inhaltsanalyse. Grundlagen und Techniken [Qualitative content analysis[END_REF]. For this report, the analysisguiding categories emerged inductively from the hypothesis, in particular, the analysis variables of teaching experience, professional background, and teacher gender. In contrast to the first item, the second item was a standardized item which was slightly adapted from [START_REF] Backe-Neuwald | Bedeutsame Geometrie in der Grundschule: Aus Sicht der Lehrerinnen und Lehrer, des Faches, des Bildungsauftrages und des Kindes[END_REF]. It included statements on reasons leading to the neglect of geometry instruction. Here, 12 statements taken from Backe-Neuwald (2000) were supplemented with three additional statements (i.e., 5.2.1, 5.2.6, 5.2.7) which were identified in the recent literature [START_REF] Clements | Teaching and learning geometry[END_REF][START_REF] Franke | Didaktik der Geometrie in der Grundschule[END_REF], with an option of writing an additional statement (i.e., 5.2.16). The in-service teachers were asked to mark the aspects that were most significant to them, and to rank five responses accordingly (e.g., 1-most significant). The item was evaluated according to the frequencies of all response options as well as according to assigned rankings. The items complemented each other; by evaluating the open-ended item, teachers could express themselves freely without having been steered in one direction by predetermined response options. Through the additional evaluation of the standardized item, the spectrum of answers was expanded.

Results

Here, I present the results concerning the two items. The first item explicitly addressed the question of whether geometry instruction is neglected in the teachers' opinion and asked them to give reasons for their answer. In total, 45.8% of the participants agreed with the statement (n = 55), 35.8% of them disagreed (n = 43), and 14.2% did not take a clear position (n = 17). Five surveys were disregarded because either no information was provided or more than one answer was given. The affirmative responses were relatively evenly distributed across gender, professional background, and teaching experience in the groups with up to 5, 20, and more than 20 years of experience. However, affirmative responses were represented above average (60%-66%) in teachers with up to 2 years and 10 years of teaching experience. Conversely, 40% to 44% of those teachers with up to 5, 20, and more than 20 years of experience did not agree with the statement.

For the qualitative analysis of the item, the responses were categorized along the chosen focal variables of teaching experience, professional background, and teacher gender and presented as a selection, since there was often considerable overlap between the answers. Looking at the teachers' justifications for their respective statements, several commonalities stand out. Geometry instruction is neglected in elementary school [...] because "for almost all teachers arithmetic is in the foreground." (female, up to 20 years, specialist) "when there is time pressure, practicing 'arithmetic' is preferred." (female, over 20 years, specialist) "the weighting of work in the textbooks is also evident. Few textbooks deal adequately with geometry topics, especially in Grades 1-4. This means that the teacher has to provide a lot of additional material and it also gives the impression that it is of less importance." (female, up to 10 years, specialist) "the RLP [curriculum] is crammed with other elementary basic knowledge and a lot of time is lost to the sustainable learning of the basic arithmetic operations." (female, up to 10 years, specialist) "many consider the topic of 'Numbers and operations' to be the absolute No. 1 and since it is so packed, geometry goes under." (female, up to 10 years, specialist)

The predominant reasons given for the neglect of geometry were time pressure (n = 15), and the importance of arithmetic (n = 22). The latter was often mentioned in connection to the mathematics standards for primary education. Furthermore, the often very time-consuming and material-intensive preparation was also frequently mentioned as a factor contributing to the neglect of geometry (n = 7), which was also brought up in connection with textbooks. The reasons given mainly focused on practical requirements and perceived constraints (i.e., time pressure, the importance of arithmetic) or convenience (i.e., intensity of preparation). A disregard for the content of geometry was rarely communicated in the statements, but did occur occasionally: "it does not have the same value as imparting basic knowledge." (female, over 20 years, specialist) Finally, in five cases the opinion was expressed, especially by those with more teaching experience, that non-specialist teachers in particular simply do not have the necessary qualifications to teach geometry confidently and appropriately:

"many (colleagues) feel insecure or perceive it as a 'gimmick'." (male, up to 20 years, specialist) "there is a lack of trained professionals." (female, over 20 years, specialist) "lateral entry employees often lack the ability of methodical approach in teaching geometry." (female, over 20 years, specialist) There are some comments stating that geometry is not neglected, but on closer inspection these display a misunderstanding of the question: they do not offer a general assessment of elementary school geometry as a whole (except in three cases), but rather refer primarily to the teacher's own practice based on school or state curricula (n = 33). Geometry instruction is not neglected in elementary school [...] because "I always meet the curriculum requirements." (female, over 20 years, specialist) "one lesson per week is a fixed geometry lesson." (female, up to 5 years, non-specialist)

The results from the standardized item of the questionnaire concerning the reasons for the neglect of geometry instruction in primary school is shown in Table 1. In addition to marking the factors influencing the implementation of geometry in school mathematics, the teachers were asked to rank the five most significant factors (i.e., statements). From this, a simple score was generated, which awarded the most important reason five points, the second most important four points, and so on. The score, as opposed to just looking at the absolute frequencies, allows a better assessment of the importance of the reasons for neglecting the teaching of geometryeven though both metrics show an almost equal ordering of the reasons. Out of 120 surveys, 117 teachers filled out this item. Additionally, not all participants ranked their answers (n = 24), so that this data was not taken into consideration for the Top 5 Score. The teachers' answers on this item predominantly confirmed the findings of the aforementioned open responses item. Accordingly, the importance of arithmetic was again the primary reason given for neglecting geometry (items 5.2.6 and 5.2.8). Item 5.2.8 was evenly distributed across the professional background with two-thirds of participants from both groups agreeing with this statement. The intensity of preparation for geometry continued to be significant, especially as evidenced by the comparatively high score (item 5.2.3). Similarly, the role of the teacher was addressed in the sense of uncertainty in the conveyance of geometry content (items 5.2.11 and 5.2.2). Item 5.2.11 was also relatively evenly distributed across professional background; 38.9% of subject specialists and 48.3% of non-specialists agreed with this statement. Thus, the issue of neglect seems to originate predominantly from temporal-practical reasons, and not from a disregard for the importance of geometry. One of the most frequent reasons that the teachers gave on item 5.2.16 (i.e., "Other: _______") was the limited material availability in schools, including both analog and digital tools. This opens up a certain area of tension, since apparently teaching material in the sense of geometry as a visual instruction is quite appreciated, but it is more likely to be made available by the school rather than to be developed as a part of the teachers' own lesson preparation. Moreover, the teachers also reported that mathematics textbooks contributed to the negative trend for the following reasons: poor coverage of geometry topics, lack of usable tasks, and lack of integration of geometry topics into the "overall mathematics".

Discussion and conclusions

In the literature, it has been postulatedoften on an insufficient empirical basisthat geometry is neglected in mathematics. This paper focuses on the question of whether there are any changes regarding the neglect of geometry teaching in the last two decades in relation to two items adapted from [START_REF] Backe-Neuwald | Bedeutsame Geometrie in der Grundschule: Aus Sicht der Lehrerinnen und Lehrer, des Faches, des Bildungsauftrages und des Kindes[END_REF]. The study results confirmed the hypothesis that despite the paradigm shift that occurred two decades ago, geometry is still insufficiently taught compared to other areas of mathematics, but positive changes were observable compared to findings reported two decades ago.

While about 80% of the participants in the study by [START_REF] Backe-Neuwald | Bedeutsame Geometrie in der Grundschule: Aus Sicht der Lehrerinnen und Lehrer, des Faches, des Bildungsauftrages und des Kindes[END_REF] agreed with the thesis that geometry instruction is neglected in elementary school, only 45.8% of participants in this study agreed with that statement. In other words, 15% of participants from the Backe-Neuwald (2000) study and 35.8% of participants in this study disagreed with the thesis. Despite the positive trend over the last two decades, the perception that geometry is neglected still exists to a high degree that cannot be disregarded. The main reason for this seems to lie predominantly in the perceived importance of arithmetic (n = 80; 86%, item 5.2.8), especially in the early school years as has been reported previously in the work of [START_REF] Backe-Neuwald | Bedeutsame Geometrie in der Grundschule: Aus Sicht der Lehrerinnen und Lehrer, des Faches, des Bildungsauftrages und des Kindes[END_REF] (n = 89; 82.4%). The fact that geometry can and must be the basis of arithmetic, both in terms of content and in the formation of mathematical teaching-learning processes [START_REF] Bauersfeld | Drei Gründe, geometrisches Denken in der Grundschule zu fördern [Three reasons to promote geometric thinking in elementary school[END_REF], is not taken into consideration by the teachers at all. Similarly, the teachers from both studies identified the preparation intensity of geometry instruction in terms of material creation and provision (item 5.2.3) as the second reason for the neglect of geometry. Both groups also emphasized that standardized materials were only available to a limited extent. Geometry lessons are guided by the visualization of different concepts and the provision and use of materials in this respect is paramount, but the reality in schools still seems to be deplorable. If improvements in this situation would make it possible to upgrade elementary school geometry, such measures should certainly be taken, because the value of continuous material use for the learning success, especially in the area of spatial imagination, is undisputed [START_REF] Franke | Didaktik der Geometrie in der Grundschule[END_REF]. It is worth noting, however, that an improved availability of materials, as an essential requirement of making changes to the teaching of geometry, would not alleviate the perceived time pressure under the impression of the excessive weight of arithmetic; yet this is a particularly pressing problem of geometry instruction for teachers. Two reasons for neglecting geometry that played a minimal role in [START_REF] Backe-Neuwald | Bedeutsame Geometrie in der Grundschule: Aus Sicht der Lehrerinnen und Lehrer, des Faches, des Bildungsauftrages und des Kindes[END_REF] with less than 10% of teachers identifying them, seem to play a greater role two decades on. Difficulties regarding performance assessment in geometry (item 5.2.5) was ranked fifth (n = 47; 50.5%), and intensity of subject preparation (item 5.2.2) was ranked sixth (n = 41; 44.1%). This may be due to curricular changes which placed extensive focus on process-oriented mathematical competencies such as problem solving and argumentation.

The present work has highlighted various aspects of geometry teaching that have direct implications for professional practice. As in the study by [START_REF] Backe-Neuwald | Bedeutsame Geometrie in der Grundschule: Aus Sicht der Lehrerinnen und Lehrer, des Faches, des Bildungsauftrages und des Kindes[END_REF], teachers felt insecure when teaching geometry (item 5.2.11). Even though this was ranked as the fourth reason in this study and the fifth reason in the study by [START_REF] Backe-Neuwald | Bedeutsame Geometrie in der Grundschule: Aus Sicht der Lehrerinnen und Lehrer, des Faches, des Bildungsauftrages und des Kindes[END_REF], the new results call for attention. The feeling of insecurity more than doubled in the last two decades, from 19.4% to 52.7%. Whereas in the study by [START_REF] Backe-Neuwald | Bedeutsame Geometrie in der Grundschule: Aus Sicht der Lehrerinnen und Lehrer, des Faches, des Bildungsauftrages und des Kindes[END_REF] the perception of inadequate previous education (item 5.2.10) was the fourth reason given (25.9%), this was reported by 22.6% of teachers, and hence was ranked 11th. The open-response items in particular bring these issues even more to light. The finding that the qualifications of many teachers, especially those who are not fully trained, not only are insufficient to teach geometry competently but also raise insecurities in teaching geometry should give cause for concern, especially since this aspect has not yet appeared as a prominent reason in the literature.

Comprehensive teacher training focusing on both content and pedagogical content knowledge is essential as well as supplementary training in order to dispel uncertainties when teaching geometry [START_REF] Jones | Making space for geometry in primary mathematics[END_REF]. Also, the collaboration between researchers and practitioners (e.g., design research, action research) may provide teachers with the needed support. We need to understand how teachers may be better prepared to play the roles that have been emphasized in the literature as well as ongoing developments [START_REF] Sinclair | New opportunities in geometry education at the primary level[END_REF]. The importance of this aspect can hardly be overemphasized, especially given the increasing proportion of not fully trained teachers. It should not be forgotten that the participating teachers only represent in-service primary teachers to a limited extent. Despite this drawback, the study results not only provide an up-to-date insight into elementary school geometry teaching but also provide new evidence for questions that have not yet been examined in the literature which call for immediate attention that should be examined by educators.

Table 1 : Reasons for neglect of the teaching of geometry Item Statement Absolute frequency (max. 93) Score Top 5

 1

	5.2.8	The arithmetic topics have to be worked through, leaving little	80	251
		time for geometry topics.
	5.2.3	Geometry lessons require a lot of preparation in terms of the	76	193
		creation and provision of materials.
	5.2.6	The arithmetic topics are more relevant for the school track	54	137
		than geometry topics.
	5.2.11	Teachers feel insecure when teaching geometry.	49	113
	5.2.5	Student performance is difficult to assess and time-	47	85
		consuming.
	5.2.2	Geometry instruction requires a lot of subject preparation.	41	105