Study on the parallelogram concept through an analysis of the Lebanese national textbooks from Grade 1 to the Grade 8 using Van Hiele’s theory

Study on the parallelogram concept through an analysis of the Lebanese national textbooks from Grade 1 to the Grade 8 using Van Hiele’s theory

دراسة حول مفهوم متوازي الأضلاع من خلال تحليل بعض الكتب المدرسية الوطنية اللبنانيّة من الصف الأول إلى الصف الثامن باستخدام نظرية فان هيل

[1]Mr. Elie Khoury إيلي الخوري

 

[2]Mrs. Haddad Manal         منال حداد

 

[3]Prof. Naim El Rouadiد.نعيم الرويدي  

 

تاريخ الإرسال: 24-11-2023               تاريخ القبول: 9-12-2023Top of Form

Abstract:

This study employs the van Hiele theory to analyze how parallelograms related tasks are handled in the Lebanese National textbooks, spanning grade 1 to 8. Previous studies have established that textbooks have a substantial impact on students’ mathematical performance, particularly when it comes to Euclidean geometry. Notably, there have been a dearth of textbooks analysis on geometry content in the Lebanese context. To address this gap, we examined 310 tasks and found that textbooks predominantly emphasize specific task types. It’s critical to strike a balance between different types of tasks within textbooks, as the nature of these tasks can profoundly impact students’ cognitive development. The results reveal a significant disparity in the distribution of van Hiele levels within the textbooks and the quasi-absence of Level 3 related tasks. This study emphasizes that in order to enhance students’ understanding of this critical geometric concept, textbooks should take more structured approach to task’s quantity and dispersion

. Keywords: Mathematics textbooks analysis, parallelogram, tasks analysis, van Hiele theory, Geometry didactics Top of FormTop of Form

ملخص:

هذه الدّراسة تستخدم نظرية فان هيل لتحليل كيفيّة التعامل مع مهام تتعلق بمتوازي الاضلاع في الكتب المدرسية الوطنية اللبنانيّة التي تغطي الصفوف من الصف الأول إلى الثامن. قد أظهرت الدّراسات السابقة أن الكتب المدرسية تؤثر بشكل كبير على أداء الطلاب في الرياضيات، ولاسيما في هندسة أقليدس. يجدر بالذكر أنه لم يُجرَ تحليل كتب مدرسية للمناهج الهندسيّة في الكتب المدرسية اللبنانية. من أجل سدّ هذا الفجوة، قمنا بفحص 310 مهمة واكتشفنا أن الكتب المدرسية تؤكد في الغالب أنواع محددة من المهام من الضروري التّوازن بين أنواع المهام المختلفة داخل الكتب المدرسية، إذ يمكن أن تؤثر طبيعة هذه المهام بشكل عميق على التنمية العقلية للطلاب. تُظهر النتائج وجود فارق كبير في توزيع مستويات فان هيل داخل الكتب المدرسية وقّلة وجود المهام المتعلقة بالمستوى الثالث. تشدد هذه الدراسة على أهميّة أن تتبع الكتب المدرسية منهجًا أكثر هيكلة في كمية المهام وتوزيعها من أجل تعزيز فهم الطلاب لهذا المفهوم الهندسي الحاسم.

الكلمات المفاتيح: تحليل كتب الرياضيات، متوازي الأضلاع، تحليل المهام، نظرية فان هيل، مناهج الهندسة

• Rationale and significance of the study:

The classroom environment incorporates a variety of educational resources as noted by Mithans, and Ivanus Gremek in their 2020 article. Within this array of resources, textbooks stand out as a primary reference point for teachers and students alike.

Textbooks are defined by Abdullah & Shin (2019) as books specifically designed and produced to communicate the national curriculum and fulfill the educational goals mandated. According to Ramadhani (2020), “Mathematics textbooks” is a term used to define materials in the field of mathematics, offering students and teacher a structured guide for facilitating the teaching and learning process. Based on the authors, they are systematically organized and carefully chosen to align with the specific objectives and learning outcomes in mathematics education.

In the field of mathematics education, textbooks utilization is essential for various reasons, one of which is the perceived reliability of this specific material (Lebrun, 2006). While textbooks are not the only instructional material that teachers use, our analysis focuses on them due to their prominent presence in the educational context (Usiskin, 2013; Johansson, 2007; Lebrun, 2006; Lebrun et al, 2002).

Johansson (2007) draws attention to the widespread use of textbooks in mathematics classrooms across the globe. The author asserts that, historically, people haven’t always understood textbooks’ true worth. To emphasize the dual role of textbooks as both representatives of the national curriculum and in teaching methods, a new level known as the “possibly implemented curriculum” has been introduced, bridging the gap between the intended curriculum and the implemented curriculum.

According to Remillard (2000), choosing a textbook serves as a useful indicator of the underlying curriculum, and that a careful analysis of textbooks provides important information into the extent to which the intended curriculum is being implemented. These findings align with a 2013 hypothesis set forth by Howson, suggesting that textbooks provide a more accurate representation of classroom practices compared to curriculum outlines. Consequently, the information gathered from textbooks analysis could offer a more comprehensive understanding of curriculum requirements as well as teaching strategies.

Even in the era of interactive whiteboards, and digital tablets, textbooks continue to hold an important place, as evidences by the numerous electronic versions offered by publishing houses (Côté, 2015).

Xin (2007) delved into the impact of different learning opportunities presented in Chinese and American Mathematics textbooks on students’ ability to solve problems. The study revealed a link between students’ ability to solve mathematical problems and the manner in which those problems were structured in the textbooks. For example, as noted by Valverde et al. (2002), the Third International Mathematics and Science Study (TIMSS) unequivocally showed that, across different nations, students’ learning outcomes are significantly correlated with the alignment of related topics in the curriculum and textbooks.

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In addition to being a valuable tool for students’ learning, textbooks also serve as significant source of information for teachers in their preparation and planning – so much that they take the role of curriculum guidelines when determining the content to be taught (Wijaya et al, 2022; Aldahmash et al., 2016; Fan et al., 2013).

First and foremost, there appears to be a perception of infallibility surrounding textbooks. In fact, according to Lebrun (2006), textbooks are perceived as a reliable source where indisputable truths are deposited. Based on Lebrun (ibid), if teachers find themselves in a situation where they do not feel entirely competent to teach the topics in the curriculum, the textbook becomes an indispensable tool that they use on daily basis, serving as source of theoretical support as well as a source of didactic information.

Sugiatno & Husna (2020), Purnomo et al. (2022) and Valverde et al. (2002) assert that classroom teaching practices can be significantly impacted by the organization and structure of mathematics textbooks. The authors contend that the design and organization of textbooks can affect the implementation of the different pedagogical approaches in mathematics classrooms, even when these textbooks are aligned with the same national curriculum, as is the case in Lebanon. Finally, as noted by Côté (2015), textbooks allow teachers to save time in lesson preparation which explain why even experiences teachers largely based their teaching on their textbooks.

Prior to the 2000s, few studies rigorously and methodically investigated teachers’ use of textbooks (Lebrun et al., 2002). However, as Côté (2015), the textbooks are presented as the foundational knowledge for teachers, and they can rely on national textbooks to standardize the education of all students in a specific country. These textbooks should outline the teaching approach as well as serve as essential guides in the educational intervention process.

Given the pivotal role textbooks play in mathematics education, it is unsurprising that they have received heightened attention within research on mathematics education over the past two decades (Fan et al., 2018; Fan, 2013; Trouche & Fan, 2018).

The importance that mathematics textbooks hold within educational community including students, teachers and educational supervisors, is the driving force behind this article. It underscores the value of textbooks in the teaching and learning of mathematics. In Lebanon, it is typical for parents or local organization to buy textbooks, guaranteeing that each student has access to the chosen one. These textbooks serve as a useful tool for completing homework assignment at home and are brought to every math lesson. As a result, the analysis of textbooks become crucial in selecting a curriculum-aligned textbook, and this study focusing on parallelograms is a part of the analysis.

 

• Literature review:

In the literature, many studies have been conducted on the analysis of textbooks, and in this section, we will highlight a few.

We begin with the thesis by Kalyankar (2019), whose strong methodological framework has notably influenced our article. Kalyankar study sought to assess the effectiveness of K-8 geometry curriculum standards and textbooks in preparing students for high school geometry using the van Hiele model as a foundational framework. This study examined mathematics textbooks and curriculum standards, including CCSSM and state-specific documents, which affect the mathematics students learning. The study compared the van Hiele levels in K-8 geometry standards and textbook instructional segments. It revealed variations between CCSSM and non-CCSSM standards, with the latter typically having lower van Hiele levels. However, higher levels we incorporated in some textbooks, particularly level 4 activities found in middle-grade textbooks, which were not highlighted in any standards documents. The results imply that aligning standards and textbooks with students’ developing spatial awareness, could improve geometrical understanding and high school geometry success rates.

The analysis was categorized into K-5 and 6-8 groups, highlighting the percentages of van Hiele levels in each of these materials. It is noteworthy, that some state standards and textbooks outperformed CCSSM in providing level 3 tasks for K-5 grades, while all textbooks for grades 6-8 included level 4 content, with Eureka Math leading the way.

In a study by Khoury, Sfeir, and Rouaidy (2023), three grade 8 Lebanese textbooks were closely examined in regards to the way they handled parallelograms using a 5-dimensional framework. The results demonstrated a deficiency of diversity in the type of tasks provided in the textbooks. Significantly missing were elements of critical thinking and problem-solving, making connections between concepts, varying degrees of complexity, open-ended questions, and tasks with real-world applications. This analysis emphasizes the need for a wider range of challenging and interactive tasks within the textbooks to promote a more stimulating and enriching learning environment.

Usiskin (2008) used a number of American textbooks as their primary resources for a micro-curricular study they conducted on quadrilaterals. Their study sought to determine whether a given quadrilateral could have an equivalent mathematical definition that is exclusive or inclusive. Furthermore, the study was grounded in the van Hiele model’s framework of geometric thinking.

In order to gain a deeper understanding of the discrepancies in the academic performance of Malaysian and South Korean students as revealed in TIMSS, Abdullah and Shin (2019) conducted a comparative analysis focusing on the content of mathematics textbooks regarding the concept of quadrilaterals in both countries. Their analysis revealed four significant differences in how quadrilaterals are taught in these two countries. These differences encompass the organization of the quadrilateral subject matter, the depth of exploration activities, the incorporation of deductive reasoning into the learning materials, and the level of challenge presented by end-of-chapter problems. In light of these findings, the authors suggest that the Division of Curriculum Development of the Malaysian Ministry of Education reevaluate the geometry content within the current mathematics textbooks to align it with curricula that has proven effective in preparing students for international examinations.

Purnomo, Shahrill, Pandansari, Susanti & Winarni (2022) investigated the cognitive challenges presented by geometry tasks in Indonesian elementary schools. Their investigation centered on a Grade 4 mathematics textbook that the Indonesian government published in 2018. The findings of their study show that about 30% of the geometry tasks in the textbooks require advanced levels of mathematical reasoning.

Gonzalez-Martin, Giraldo & Souto (2013) explored the integration of real numbers in secondary education. Using Chevallard’s Anthropological Theory of Didactics, the authors thoroughly analyzed Brazilian textbooks. Their findings showed that the introduction of irrational numbers heavily relies on the decimal representation of numbers, with a lack of explicit instruction regarding the mathematical foundations for constructing the real number system.

Sievert, den Ham, Niedermeyer & Heinze (2019) focused on investigating the impact of textbooks of the adaptive expertise development of primary students in arithmetic. By using a multilevel analysis, the authors investigated the quality of textbooks and its impact on their-graders’ development of adaptive expertise in multi-digit addition and subtraction. This important study highlights how textbooks have a significant long-term impact on students’ adaptive expertise.

Although other instructional materials are employed in classrooms, they are not as widely utilized as textbooks as noted by Glacin (2018). The results of the study show that teachers believe that the quality of textbooks’ tasks in the most critical factor when selecting one. The author highlights the indispensable role of mathematics textbooks in education, as students primarily use textbook tasks for practice. It is essential for textbooks to offer a wide variety of tasks because the nature of tasks can have a big influence on students’ cognitive processes.

Our particular focus is how the concept of parallelograms is presented in a selected 1-8 Lebanese textbooks collection. We aim to investigate the tasks assigned to students in order to organize and consolidate their knowledge. In this study, the primary emphasis is placed on the nature of the tasks given to students using van Hiele theory as a foundational framework.

 

• Cognitive demand of textbooks:

The present study places significant emphasis on mathematical tasks. According to the research findings, students frequently turn to textbooks as a source of tasks (Pepin & Haggarty, 2001; Rezat, 2006; Jukić Matić & Glasnovic Gracin; 2016). In line with Christiansen and Wealther (1986), tasks are defined as prompts and challenges presented by the teacher or textbook, and sometimes initiated by the students themselves, and they include particular goals that need to be completed. According the Skovsmose (2001), the teacher serves as the catalyst by “encouraging” the students to engage with the task, and the students’ response to this invitation becomes their subsequent activity. Ponte & Marques (2011) stresses that, tasks are external, whereas activities are internal to the student.

In a broad sense, tasks within textbooks are seen as ways to initiate students’ activity, as noted by Mason and Johnston-Wilder (2004) and as opportunities for learning mathematics as emphasized by Sullivan et al. (2013). As such, it’s critical to recognize that the nature of these tasks hold the potential to influence and direct students’ cognitive processes. These tasks can either narrow or expand students’ perspective on the subject matter they are studying, significantly influencing their conceptual understanding of specific mathematical concepts as noted by Henningsen & Stein (1997).

Gracin (2018) highlighted how important it is that tasks are valuable and well-structured in mathematics textbooks and other educational resources. Given the significant influence of textbook tasks and examples on mathematics education, it is perfectly reasonable to inquire about the nature and requirements of these tasks. We should investigate whether they contribute to enhancing mathematical comprehension on the concept of parallelograms and to what extend they can contribute to building a good concept-image on the concept of parallelogram.

There is limited amount of information regarding how these factors are considered in Lebanon when addressing the concept of parallelograms, spanning from grade 1 to grade 8. In this study, the textbook analysis will be conducted through the lens of van Hiele model. Specifically, we investigate the concept of parallelograms, in Euclidean geometry, spanning from grade 1 to grade 8. This scope includes geometry topics related to geometric shapes and their relationships; it does not include measurements, analytical geometry, or area calculations.

• Van Hiele theory:

To achieve our goal, the theory of levels of thought in geometry, as defined by van Hiele and van Hiele-Gledof (1958), was used. Within the realm of geometry, it is widely acknowledged, that van Hiele’s model is the most well-defined model for understanding students’ thinking levels (Alex & Mammen, 2016; Rizki, Frentika & Wijaya, 2018).

As mentioned by Braconne-Michoux (2014), this theory of how geometry thinking develops, separates five levels, hierarchically connected to on another. The van Hiele couple emphasized the need for student to master each level of thought successively, without “skipping” any, to attain the higher level of geometric thinking, which is mastery of proof and hypothetical-deductive reasoning within an axiomatic system.

Every level of thought has its own language, and two people operating at distinct levels, even if they are using the same vocabulary, they cannot understand each other. What is implicit at one level becomes explicit at a subsequent level. The transition from one level to the next is facilitated by learning conditions that are defined in terms that today evoke the ideas of the proximal zone of development, as defined by Vygotsky and adopted by Vergnaud (2000).

The five levels can be summarized as follows:

• Level 1[4]: The level of Identification – Visualization (L1). Figures are identified based on the overall appearance or “judged by their look” (van Hiele, 1958).
• Level 2: The level of analysis (L2). Although the properties of figures are known, they are not organized in any particular order. At this level, rectangles are not considered as a particular type of parallelogram, although students can identify a figure from a coded diagram of a freehand drawing.
• Level 3: The level of Informal Deduction (L3). At this level, the properties of figures are arranged, and they are deduced from one another but are not structured within an axiomatic system. Specifically, the inclusion of figures (quadrilaterals, triangles) is included in this level.
• Level 4: The level of Formal Deduction (L4). This level of geometric thinking development is characterized by the roles of axioms, definitions, direct or inverse theorems. At this level, a student can compose a formal proof.
• Level 5: The level of Rigor (L5). This level has been described as the level at which a student comprehends the various axiomatic systems in geometry. But later on, P.M van Hiele himself questioned whether such a level even existed and, more importantly, whether it was useful in studying the development of geometric thinking in a student from the early primary school to the end of secondary school. It’s important to note that the fifth level extends beyond the scope of geometry teaching in primary and secondary schools. As a result, our focus will be on the first four levels. Top of Form

Although van Hiele model mainly focuses on analyzing students’ cognitive levels and thought processes related to specific geometric concepts, rather than textbook tasks, it is important to note that tasks are a major factor in encouraging students’ engagement and activity (Gutierrez, Jaime & Fortuny,1991; Kalyankar, 2019). As a result, applying this theoretical framework to task analysis provide valuable insights into both student activity and their thought processes regarding the subject matter.

According the van Hiele Geldof, as mentioned in Haviger & Vojkůvková (2014), it takes approximately 20 lessons to progress from L1 to L2 and 50 lessons to progress from L2 to L3. This suggests that the rate of development through the levels varies, especially in early stages. It’s reasonable to assume that progressing to L4 from L3 would require even more lessons. Building upon many studies (Fuys et al., 1988; Pace, 1991; Abdullah & Zakariya, 2013), it becomes evident that it is not feasible to teach students two levels concurrently within the same grade.

• Methods:

Various approaches to textbook analysis exist, but for our study, we will adopt one specific method. This prevalent method, as highlighted by Van Dormolen (1986), namely “A priori textual analysis”, entails the examination of a text with the perspective of assessing its potential as an instructional tool. This study employs a content analysis technique based on document analysis.

Regarding the topic, we chose the concept of parallelograms considering its fundamental role in geometry curriculum in the Lebanese context and also in other contexts (NTCM, 2000). This choice aligns with the Lebanese curriculum, since the trapezoids are introduced in conjunction with the midpoint theorem. Throughout the study, the term “task” refers to problem-situations found in textbooks, regardless of whether they include provided solutions. These tasks can take the format of an exercise, worked examples, revision tasks, and other types of problems. This analysis focused also on the English versions of the textbooks, as English serves as the common language among the authors.

Our primary focus is not the comparative analysis of textbooks. Instead, we narrow our scope to textbooks from a specific collection, namely the national Lebanese textbook series published by the Center for Education Research and Development (CERD). The collection is titled “Building up mathematics” published in 1999. The national textbook serves as a comprehensive, standardized resource, offering students and teachers a valuable tool for enhancing for teaching and learning mathematics. We consider the students’ copies of the textbooks, while the teachers’ guides provide additional evidence to support our assessments of the assigned van Hiele levels for each task.

In each grade, we have carefully chosen the chapters directly linked to the concept of parallelograms. We have also broadened our analysis to include parallelogram exercises and problems within the Euclidean geometry topic, even when they are woven into other chapters.

Within grade 1, the selected chapters comprise: Geometric shapes, Draw with a ruler, Superposition of figures. In the context of grade 2, the chapters we’ve chosen are: Measurements of lengths (The centimeter), Understanding a statement, Plane figures, Shapes and tracings. For grade 3, our focus lied on the following chapters: Following visual instructions, Square and rectangle, Symmetry.  In the grade 4, we have singled out the chapters: Quadrilaterals, Symmetry with respect to an axis. In grade 5, our attention was directed towards the chapter: Characteristic of quadrilaterals. For grade 6, we’ve opted to analyze the chapters: Lines and circles, Central symmetry. Within grade 7, we’ve specifically pinpointed the chapters: Translations, Angles and lines. Grade 8 involves the examination of the chapters: Square, Rectangle, Rhombus, Parallelogram.

Considering the stablished significance of van Hiele model in the realm of geometry education, specifically in relation to parallelograms, and the critical role mathematics textbooks play in shaping both teacher instruction and student learning, this article seeks to achieve the following objectives:

• Evaluate the distribution of van Hiele levels in tasks related to the concept of parallelograms across grade levels in Lebanese national textbooks for grade 1-8.
• Examine whether Lebanese national textbooks for a particular grade covers tasks that span multiple van Hiele levels related to the concept of parallelograms.

Every textbook was meticulously examined on a page-by-page basis. Predominantly, the tasks chosen were from exercises within sections labeled “Exercises, self-evaluation, and problems”. We also included worked examples found in the “subject” section of the textbooks for our analysis. Notably, the “just for fun” and “activity” sections were excluded from our selection criteria. We continued to concentrate only on coding tasks related to the parallelogram concept.

Three researchers – the authors – were engaged in the meticulous task of coding textbook content. One researcher is a professor of education, specializing in curriculum and textbooks design in mathematics. Another team member, is a Ph.D student in education, brings extensive experience in mathematics coordination, teaching, and textbook development. The third researcher, a Master’s in education student, has extensive a strong teaching background and significant involvement in crafting mathematics assessments for national exams.

The researchers underwent rigorous training in van Hiele theory, prior to beginning the coding process, including an in-depth exploration of thinking process and corresponding responses across all levels, using the substantial corpus of work by van Hieles and other well-know researchers in the field. This comprehensive training has enabled the team to enlightening conversations about hypothetical tasks and student responses, which made it possible for each researcher to determine the appropriate van Hiele level to individual thoughts or responses. In cases where disagreements arose, the researchers engaged in a thorough collective review of the theory and referred to original research examples until a consensus was achieved, ensuring a shared understanding of both textbooks’ tasks and van Hiele levels. This rigorous collaborative procedure significantly enhances the reliability of the coding process.

Reliability in coding is paramount, encompassing three key areas: accuracy, reproducibility, and stability. Stability ensures consistent coding over a range of time points, achieved by sticking to a set of coding scheme and completing coding tasks both before and after learning more about van Hiele levels. Reproducibility, also known as inter-coder reliability, gauges how well-coding categories are defined when the three coders work together. Finally, accuracy assesses how well a procedure aligns with established standards.

Tasks at L1 involve recognizing figure based on the general appearance (Figure 1), L2 tasks require analyzing the properties of figures (Figure 2), L3 tasks involve information deduction, such as classifying figures hierarchically or understand defining properties, with some experience in proofs (Figure 3), L4 tasks require formal deduction from previously learned geometric facts to prove unknown facts in geometry (Figure 4).

Figure 1:Task at L1 in Grade 1 textbook on page 52.

Figure 2:Task at L2 in Grade 8 textbook on page 41.

Figure 3: Task at L3 in Grade 8 textbook on page 57

Figure 4:Task at L4 in Grade 8 textbook on page 74

It’s noteworthy to mention that the task in Figure 4 initially seems like a L1 tasks since it involves sorting shapes based on their appearance. However, after looking through the teachers’ guide, it becomes clear that students should consider square as a special type of rectangles, requiring L3 thinking to solve this task. As a result, it is coded as L3. As for the task in Figure 5 labeled as L4, the use of the verb “justify” is used in an ambiguous way at first. However, after referring to the teachers’ guide, it becomes evident that the response requires a mathematical proof, necessitating a L4 coding.

• Results:

A total of 310 tasks were identified during the analysis across all textbooks, covering grade 1 through 8.

Table 1: Number and percentage of all van Hiele levels across Mathematics textbook for grade 1 to 8

Van Hiele levels	Number of tasks in 1-8 textbooks	Percentage of tasks in 1-8 textbooks
L1	94	30.32%
L2	90	29.03%
L3	5	1.61%
L4	121	39.04%
Total	310	100

Table 1 reveals a breakdown of tasks across van Hiele levels as follows: L1 comprises a total of 94 tasks, representing 30.32% of the total; L2 encompasses 90 tasks, accounting for 29.03%; L3 contains a smaller emphasis, with just 5 tasks, equivalent to 1.61%; L4 displays a substantial emphasis, featuring 121 tasks, making up 39.04% of the total. It become evident, based on the analysis, that the importance is given to L1, 2 and 4, while L3 receives relatively limited attention.

Table 2: Number and percentage of tasks linked to the concept of parallelograms in 1-8 textbooks

Grade level	Number of tasks	Percentage of tasks
1	14	4.51%
2	17	5.48%
3	12	3.87%
4	23	7.42%
5	22	7.09%
6	6	1.94%
7	1	0.33%
8	215	69.36%
Total	310	100

In Grade 1, there are a total of 14 tasks constituting 4.51% of the total tasks analyzed.  In Grade 2, there are 17 tasks, accounting for 5.48% of the total. Grade 3 contains 12 tasks, representing 3.87% of the total tasks analyzed. Grade 4 includes 23 tasks related to parallelograms, making up 7.42% of the total. Grade 5 feature 22 tasks, constituting 7.09% of the total tasks analyzed. Grade 6 comprises 6 tasks amounting to 1.94% of the total. Grade 7 has only 1 task related to parallelograms, which is 0.33% of the total tasks. Finally, in Grade 8, there is significant emphasis on the concept of parallelograms, with 215 tasks, making up the majority at 69.36%.

These findings reveal, as students move from lower to upper grades, a gradual increase in the number of tasks related to parallelograms, with a significant emphasis on the concept in Grade 8. Interestingly, there seems to be a quasi-gap in the tasks related to parallelograms in Grades 6 and 7, where the number of tasks is relatively limited compared to the surrounding grade levels.

Table 3: The number and percentage of tasks corresponding to various van Hiele levels in every grade

Levels Grades	L1	L2	L3	L4	Total
1	14	–	–	–	14
	4.51%				4.51%
2	17	–	–	–	17
	5.48%				5.48%
3	12	–	–	–	12
	3.87%				3.87%
4	23	–	–	–	23
	7.42%				7.42%
5	22	–	–	–	22
	7.09%				7.09%
6	–	6	–	–	6
		1.94%			1.94%
7	–	1	–	–	1
		0.33%			0.33%
8	6	83	5	121	215
	1.94%	26.77%	1.61%	39.03%	69.36%
Total	94	90	5	121	310
	30.32%	29.03%	1.61%	39.04%	100%

The analysis reveals a noteworthy pattern in the distribution of van Hiele levels across different grades. Grades 1 to 5 predominantly focus on L1 tasks, with minimal inclusion of L2 tasks in grades 6 and 7. However, a significant shift occurs in grade 8, where tasks encompass L2, 3 and 4 with a limited representation of L3 tasks.

• Discussion:

The results demonstrate a significant discrepancy in the distribution of van Hiele levels within the Lebanese National textbooks, particularly in relation to the concept of parallelograms. From grades 1 to 5, L1 tasks are the main focus for students, offering them foundational knowledge and skills related to parallelograms. However, fewer L2 tasks are introduced in grades 6 and 7, suggesting a slight shift towards more advanced levels. In stark contrast, grade 8 introduces tasks across L2, 3 and 4, which presents a challenging scenario for students (Fuys et al., 1988; Pace, 1991; Abdullah & Zakariya, 2013). The most concerning aspect is the scarcity of L3 tasks, a critical stage for students to transition to a more axiomatic geometry Braconne-Michoux (2014).

This uneven distribution raises significant concerns regarding how well students understand the parallelograms concept. Grade 8 students, after limited exposure in previous grades, are suddenly confronted with tasks that require multiple cognitive levels. This abrupt progression could make it more difficult for them to fully comprehend the subject in hand. These findings suggest that many students may remain at L1 or make minimal progress to L2, which may indicate that they may have a weak concept-image of parallelograms.

Consequently, it is necessary to investigate how textbooks and instructional strategies influence students’ comprehension of this important geometric concept. By placing greater emphasis on higher levels, especially level 3, and encouraging a deeper understanding of geometric figures, teachers can better support students on their mathematical journey, ultimately leading to enhance learning outcomes in geometry and mathematical proof.

• Conclusion:

This paper showcases an analysis of mathematics textbooks using an analytical framework rooted in didactic principles. The focus of this study centers on the prescribed tasks within the Lebanese national textbooks from grade 1 till grade 8, with a framework based on the van Hiele model of geometric thoughts. The results underscore the prevalence of certain levels over other, shedding light on the instructional dynamics at work in these learning materials.

This study recommends, on the one hand, broadening the scope of tasks in terms of van Hiele levels. Classifying tasks based on each level within van Hiele model would be beneficial. Such classification, could be utilized by textbooks authors in Lebanon to enhance teachers’ awareness of the task types they are presenting. They should also reevaluate the distribution of tasks within grade levels, considering a more balanced and thoughtful allocation tasks across various levels to enhance students’ learning experience.

On the other hand, teachers should consider different factors, when evaluating textbooks for adoption and organizing their lesson, especially within the context of van Hiele levels. Teachers should make sure that the body of tasks correspond with the appropriate van Hiele levels and cover a wide range of them.

Finally, it will be interesting to observe the practical use of these textbooks by both students and teachers in future studies. Of particular interest is the extent to which the learning opportunities presented by these textbooks translate into classroom activities, and ultimately, affect students’ learning outcomes.

• References:
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• Abdullah, A., & Zakaria, E. (2013). The Effects of Van Hiele’s Phases of Learning Geometry on Students’ Degree of Acquisition of Van Hiele Levels. Procedia – Social and Behavioral Sciences, 102, 251–266. https://doi.org/10.1016/j.sbspro.2013.10.740
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[1] – طالب دكتوراه في تدريس الرياضيات، جامعة القديس يوسف  (USJ)بيروت.

PHD Student in Maths Didactics, University Saint Joseph (USJ), Beirut, Lebanon E-Mail: Eliekhoury14@hotmail.com

[2] – طالبة ماجستير في تدريس الرياضيات، الجامعة اللبنانية، بيروت، لبنان.

Masters student in Math’s didactics, Lebanese university, Beirut, Lebanon, E-Mail:  manal.c.haddad.85@gmail.com

[3] – جامعة القديس يوسف بيروت وجامعة البلمند الكورة، لبنان.

Saint Joseph  University  (USJ) Beirut, University Of Balamand    (AOB) , Koura,  Lebanon Email: Naim.rouadi@gmail.com

[4] Van Hiele levels were originally numbered from 0 to 4. Nevertheless, for practical reasons, we will number from 1 to 5, as this is the standard practice employed in most publications related to this theory.