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Young children's concepts of geometric shapes
Abstract and Figures
The aim of the mathematical publication „Young Children‘s Conceptions of Geometric Shapes“ was to present findings of the three years long research project supported by the Scientific grant agency of the Ministry of Education of Slovak republic and the Slovak Academy of Sciences VEGA 1/0440/15 named „Geometric conceptions and misconceptions of children of pre-school and school age“, which focused on determining the conceptions of children of pre-school and younger school age about planar geometric shapes.
The research was conducted in common slovak socio-cultural environment during 2015-2017, with 485 children aged 4-11 participating in total. The selection was available and located in north and north-east Slovakia. Considering the broad age range of the respondents, we divided the research sample into three subsamples (60 pre-school children, 80 first-graders and 345 fourth-graders) and used our own, age-appropriate research tool fir each of the groups. We verified the appropriateness and objectivity of the research tools statistically.
By conducting individual semi-structured interviews, at which we used models of shapes, our own graphic hand-outs and interview scripts, we examined the ability of children to name a planar geometric shape (circle, triangle, square, oblong), identify a shape given its name, distinguish models and non-models of a given shape. With students, we also examined the ability to create a model of a planar geometric shape in a square grid by pre-stated rules. In the test of knowledge in the oldest segment of our research sample, we also examined whether the children know significant elements and properties of the geometric shapes in question. The research included direct and indirect observation, the interviews were transcribed and assessed quantitatively using mathematical statistical methods, but also qualitatively, where we observed and evaluated verbal and non-verbal expressions of children and compared them with existing characteristics of the van Hiele levels of geometric thinking.
Children’s conceptions about geometric shapes exactly matched the van Hiele level of visualisation, were strongly influenced by prototypes. Occasionally, in some of the children’s reasoning we observed signs of the van Hiele level of analysis. The stability of the children’s geometric preconceptions is high. We identified holistic perception of geometric shapes and their comparison with visual prototypes virtually in all three age groups. Active usage of the names (terminology) was more difficult as passive usage, and the success rate of identification of individual shapes was at a comparable level. The size, form and position of the shapes had significant influence on the success rate as well as the shape’s variability in form and the children’s experience level. According to our findings, each child may manifest signs of different levels for different shapes. Some shapes prove to be more difficult for individuals, some easier, some they are only able to perceive holistically and in some they take notice of certain significant elements. Boundaries between individual van Hiele levels are blurred, even non-existent. Therefore, considering our research findings, we cannot understand the van Hiele levels as isolated, or disjoint, phases.
Concerning studying geometric thinking of children of pre-school and younger school age, this publication is the first complex and comprehensive work in Slovakia providing real empirical data.
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... As Kožuchová and the team of authors [3] state in their publication Curriculum of Primary Education, pupils at the primary level (ISCED 1) should acquire new knowledge and skills especially via their own activities and experiences. They should learn to express the basics of their literacy, including mathematical literacy, as much as possible and communicate it to develop it further [4]. However, teaching mathematics at the primary level is often completed by passively listening to the interpretation of the new curriculum without approaching the practical applicability of the acquired knowledge and skills. ...
The paper deals with the application of concept mapping in teaching mathematics at the primary level of education (ISCED 1), as well as the experimental verification of its qualitative contribution to the process of education. The authors present results obtained in a sample survey aimed at monitoring possibilities of influencing pupils’ relationship to teaching and mastery of mathematics using digital concept maps within the process of pedagogical intervention. The authors tested the degree to which digital concept maps influence changes in the pupils’ opinions and attitudes to mathematics (subject popularity, interest in the subject, easiness of understanding, fear and apprehension of mathematics) while explaining and fixating the curriculum of selected mathematics topics in the third grade of a primary school. The results of the experiment did not confirm that teaching supported by concept maps contributes to reducing pupils’ negative attitudes towards subjects, particularly in mathematics at the primary level (ISCED 1). Due to the pedagogical intervention of concept mapping, no statistically significant changes occurred in the pupils’ attitudes to the observed factors of mathematics. However, the authors have come up with research findings that may be notable in the field of branch didactics of mathematics and digital technologies.
... For example, pre-school children do not have a problem designating a prototypical triangle (an equilateral triangle with one horizontal side) as a triangle, while they will hesitate in the case of an obtuse triangle (Clements et al., 1999;Tirosh & Tsamir et al., 2011). Pre-school children also usually recognise a prototypical square in a standard position with horizontal and vertical sides (Budínová, 2017a(Budínová, , 2017bGunčaga et al., 2017;Žilková, 2018). ...
Understanding basic geometric concepts is important for the development of students’ thinking and geometric imagination, both of which facilitate their progress in mathematics. Factors that may affect students’ understanding include the teacher of mathematics (in particular his/her teaching style) and the textbooks used in mathematics lessons. In our research, we focused on the conceptual understanding of trapezoids among Czech students at the end of the 2nd level of education (ISCED 2). As part of the framework for testing the understanding of geometrical concepts, we assigned a task in which students had to choose trapezoids from six figures (four trapezoids and two other quadrilaterals). In total, 437 students from the 9th grade of lower secondary schools and corresponding years of grammar schools participated in the test. The gathered data were subjected to a qualitative analysis. We found that less than half of the students commanded an adequate conceptual understanding of trapezoids. It turned out that some students did not recognise the non-model of a trapezoid, or vice versa, considering a trapezoid in an untypical position to be a non-model. Our research was supplemented by analysis of geometry textbooks usually used in the Czech Republic which revealed that the trapezoid is predominantly presented in a prototypical position. Moreover, we investigated how pre-service teachers define a trapezoid. Finally, we present some recommendations for mathematics teaching and pre-service teacher training that could help.
The monograph focuses on inquiry-oriented teaching in the higher teacher education preparation of future primary teachers in technical education. The aim of the monograph is to present the results of the research carried out within the project VEGA grant 1/0033/22 Inquiry oriented teaching in mathematics, sciences, and technical education and KEGA grant 026UK-4/2022 Concept constructionism and augmented reality in STEM education (CEPENSAR).
The aim of the monograph is to present the results of the research carried out within the project VEGA 1/0033/22 Inquiry-oriented teaching in mathematics, sciences, and technical education with students of the study programme Teacher Training for Primary Education in mathematical subjects.
Research shows that children aged 3 to 6 years do not fully grasp the concept of geometric shapes. This paper aims to examine children's intuitive knowledge of triangles and squares. We analysed the effects of distractors on identification and the neglected properties of (non)examples. The purpose of the study was to establish the developmental path in the identification of shapes. It was operationalized by determining types of interfering distractors in shape recognition and properties neglected. The data obtained from individual interviews were processed by the method of statistical and descriptive qualitative analysis. A classification was made of distractors and properties of non-examples affecting identification.
Understanding of space and geometric thinking are some of the important human abilities useful for everyday life. The development of spatial sense and the development of geometric thinking are inseparable from learning and teaching geometry. According to van Hiele's theory, students develop their geometric thinking through appropriate activities and progress through the levels. The paper presents several activities that can improve the development of spatial sense and geometric thinking of elementary school students.
The scientific monograph presents a theoretical-empirical view on the topic of motivation to learn in the school environment. We have chosen as the main approach an autonomy supporting teache from the Theory of self-determination. We present our several research findings from the field of assessing the style of motivation to learn in lower secondary education and students in university programs pre-primary and primary pedagogy. Due to the lack of direct data from the area of measuring motivation in pupils at the pre-primary and primary level, we have extended our analysis with data from the International Measurement of Reading (PIRLS), Mathematical and Science Literacy (TIMSS). We assumed that the nature of motivation to learn is directly linked to the results of the learning process. The concept of literacy is comprehensive for linking with the concept of motivation. In the monograph, we brings a concept of change based on the principles of personalized learning, enriched by an approach aimed at building a culture of environment focused on innovation as a starting point for negative research conclusions. The monograph focuses mainly on the role of the future teacher, whom he perceives as crucial person in the process of introducing any new positive changes.
Araştırmanın amacı okul öncesi dönem çocuklarının geometrik şekil algılarının incelenmesidir. Nitel araştırma desenlerinden olgubilim deseninde yürütülen araştırmanın çalışma grubunu 2019-2020 eğitim-öğretim döneminde Denizli ili merkezindeki MEB’e bağlı anasınıflarına ve bağımsız anaokulları devam eden 48-66 ay arasındaki 100 çocuk oluşturmaktadır. Verilerin toplanmasında Demografik Bilgi Formu ve Geometrik Şekilleri Tanıma Testi kullanılmıştır. Verilerin analizinde çocukların Geometrik Şekilleri Tanıma Testi’ne verdikleri cevaplar kod ve kategorilere ayrılarak incelenmiştir. Araştırmanın sonucunda çocukların vermiş oldukları cevaplardan “görsel cevaplar”, “niteliksel cevaplar”, “görsel ve niteliksel cevaplar”, “anlamlı olmayan cevaplar” kategorileri ve “Bilmiyorum” cevabı elde edilmiştir. Kategorilerin yaşa göre incelenmesi sonucunda 4 yaş çocuklarının geometrik şekilleri tanımada daha çok görsel cevaplar verdikleri, 5 yaş çocuklarının ise daha çok niteliksel cevaplar verdikleri görülmüştür. Görsel ve niteliksel cevaplar kategorisi için dikdörtgen ve daire şekillerini tanıma boyutlarında 5 yaş çocukları daha çok görsel ve niteliksel cevaplar verirken, 4 yaş çocuklarının daha çok anlamlı olmayan cevaplar verdikleri sonucuna ulaşılmıştır.
Zusammenfassung
Zu den ersten geometrischen Begriffen, die Kinder bereits im Elementar- und Primarbereich lernen, zählen u. a. Viereck, Rechteck und Quadrat. Studien zeigen, dass Lernende bereits früh individuelle Vorstellungen, sog. individuelle Begriffskonzepte, zu diesen Begriffen aufbauen. Zwar wird die Entwicklung von Begriffsverständnis in verschiedenen mathematikdidaktischen Stufenmodellen dargestellt, diese sind jedoch generisch und beschreiben nicht explizit die Entwicklung der ersten individuellen Begriffskonzepte von Lernenden zu Viereck, Rechteck und Quadrat. Aus empirischer Sicht liegen verschiedene Studien vor, die einzelne Aspekte der individuellen Begriffskonzepte von Lernenden unterschiedlicher Altersgruppen zu diesen Begriffen ausleuchten. Um Begriffsbildungsprozesse aus empirischer Sicht detaillierter entlang der jeweils vorherrschenden individuellen Begriffskonzepte zu beschreiben, fehlen insbesondere Studien in der Grundschule, die alle vier Klassenstufen betrachten und dabei differenzierte Erkenntnisse zu verschiedenen theoretischen Indikatoren des Begriffsverständnisses liefern. Daher geht die vorliegende Studie der Frage nach, welches Verständnis der Begriffe Viereck, Rechteck und Quadrat Schülerinnen und Schüler der Jahrgangsstufen 1, 2, 3 und 4 zeigen. Dazu wurde eine Quasi-Längsschnittstudie mit N = 456 Grundschulkindern (ca. 100 pro Jahrgangsstufe) durchgeführt. Die Ergebnisse geben detaillierte Einblicke in die individuellen Begriffskonzepte der Lernenden und zeigen, dass Lernende zunehmend Eigenschaften der Figuren berücksichtigen, jedoch individuelle Begriffskonzepte über lange Zeit auch prototypisch geprägt sind. Implikationen dieser Ergebnisse für Forschung und Praxis werden diskutiert.
Alternatívne predstavy a ich základné charakteristiky (Vymedzenie pojmov, Základné znaky alternatívnych predstáv, Typické znaky prvotných predstáv v prírodovednom poznávaní)
Vplyv alternatívnych predstáv na proces učenia (Asimilačná teória vzniku poznatkov, Kognitívny vývin jedinca podľa teórie konštruktivizmu, Konštruktivistické vyučovanie)
Metódy rozpoznávania alternatívnych predstáv (Výskumné metódy rozpoznávania alternatívnych predstáv, Metódy rozpoznávania alternatívnych predstáv v školskej praxi)
Koncepčné učenie (Podstata a úloha kognitívneho konfliktu v koncepčnom učení. (Re)Konštrukcia predstáv, Zhodnotenie (re)konštruovaných predstáv, Bariéry koncepčných zmien, Príklady stratégií učenia podporujúcich koncepčné zmeny, Učebné osnovy pri koncepčnom učení)
Príklady alternatívnych predstáv ovplyvňujúcich poznávanie fyziky (Vedecké poznanie, veda; Sila a pohyb; Elektrické javy; Magnetické javy; Svetelné javy; Zvuk; Termika, tepelné javy; Časticové zloženie látok, štruktúra a vlastnosti látok rôznych skupenstiev)
The purpose of this study was to explore preschool children's conceptual understanding of geometric shapes, the square in particular. There were a total of 115 children, 61 girls and 54 boys, from state preschool education programs, who participated in the study. The data were collected in two semesters through interviews in a one-on-one setting, where the researchers administered a paper-pencil test to the participants. The test included six questions. One question asked children to draw a square, one question asked to select the square among three other geometric shapes, three questions asked to differentiate the square among five to seven geometric shapes which were printed in rotated directions and in various fonts and sizes and one question asked to identify a picture of a square-like real life object among a selection of pictures. The findings showed that 65% of children were able to draw a square accurately, and 77% of children were able to identify a picture of a square-like object. Approximately 69% were able to differentiate three squares in different sizes among five geometric shapes, while 27% of the remaining were not able to identify the square in smaller sizes. Approximately 79% in one task and 56% in another task were unsuccessful in identifying squares in rotated directions. Moreover, there was no gender difference in the test between boys and girls. Findings were interpreted linking to Duval's theory, van Hiele's theory, Prototype theory and Simon's task design model.
Domestic and foreign studies have confirmed that pre-service teachers in primary education have various misconceptions about geometric ideas. In this paper we analyze the results of research about the misconceptions in quadrilaterals and its potential causes in Slovak students. The aim of the research was to find out how students perceive quadrilaterals, which properties of quadrilaterals are the most problematic and which inclusive relationships among sets of quadrilaterals are not acceptable for the students. We used Van Hiele theory about development of geometric thinking and implicative statistical analysis in the analysis and interpretation of the research results.
Geometric shapes are one of the primary subjects in mathematics education during early childhood. Even before entering formal schooling, children will have some basic information about geometric shapes through their everyday experiences. Some of this early information about geometric shapes might be erroneous, which might negatively impact children’s further understanding of geometric shapes. For that reason, exploring how children recognize and classify geometric shapes and the criteria they use for classification is critical in determining the content of early mathematics education. Van Hiele’s theory about the development of geometrical thinking is regarded as one of the basis theories concerning children’s classification of geometric shapes. However, recently there seems to be some arguments put forward claiming that van Hiele’s theory is insufficient in explaining development of geometric thinking in early childhood. Thus, this study was conducted to determine 3 to 6 year-old-preschoolers’ and primary 1st grade and 4th grade students’ recognition levels of geometric shapes and the criteria they use to distinguish one group of shape from the other. To this end, individual interviews were conducted with 150 children. Four classification tasks (triangle, rectangle, square and circle) were used in this study as data collection tool. The data collection tool used in this study was developed by the researchers based on the previous studies. Each classification task consisted of classifying 12 shapes, which were outlined on an A4 size paper. Results of the study revealed that while young children were successful in recognizing typical examples of the geometric shapes, they were not successful enough in recognizing the atypical examples (such as shapes with different size, orientation, aspect ratio skewness). It was also determined that younger children were paying attention mostly to visual attributes of the shapes in the process of classification. In contrast, older children were found to pay attention both to visual and property attributes or only to property attributes. Thus, results of the present study hand support to recent criticism of van Hiele’s theory
How can we improve geometry instruction at the preschool and primary levels? To answer that question, I conducted research to analyze young children's understanding of the geometric concepts of triangle and rectangle and to determine patterns in the development of this understanding from ages 3 through 6. The research suggests that early childhood educators need to rethink the way that basic shapes are introduced to young children. Since a basic understanding of shapes is essential to a future study of geometry, teachers need to focus on how best to help children develop that initial understanding of shape categories. After a brief explanation of the research, specific ways to present developmentally appropriate activities designed to enhance children's understanding of basic shapes are discussed.
This study explored 5-6 year-old children's conceptual understanding of one geometric shape, the triangle. It focused on whether children could draw a triangle from memory, and identify triangles of different types, sizes, and orientations. The data were collected from 82 children attending state preschool programs through a one-on-one interview, during which children completed five paper-pencil tasks. Findings showed that the majority of the children (93%-96%) successfully identified a prototyped triangle. Approximately half of the children experienced difficulties in identifying triangles of different sizes, types, and orientations. The most difficult area was identifying types of triangles, where an isosceles triangle and a right triangle were presented, followed by identifying triangles in different orientations, specifically flipped and rotated ones. Children appeared to identify and classify the triangle relying on the visual prototype. These findings provide support for the Prototyped Theory, van Hiele theory, and for works by Clement and colleagues.
