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Studying geometry allows students to explore and analyze shapes and spaces. It also allows for students to make those real life connections for greater understanding. It will also, allow students to be more abstract as well as grasp a deeper level of thinking which benefit them in the long run. Van Hiele levels coincides with a child’s developmental stage. Level 0 is where students use visualization to recognize shapes. Level 1 is where the student analyzes and compares shapes. During the 3rd level, students analyze informal arguments and begin to develop geometric truths. The last level is the highest level where students compare and contrast different axiomatic systems of geometry. These levels will give each student the skillset to advance throughout their education.

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Studying geometry allows students to explore and analyze shapes and spaces. It also allows for students to make those real life connections for greater understanding. It will also, allow students to be more abstract as well as grasp a deeper level of thinking which benefit them in the long run. Van Hiele levels coincides with a child’s developmental stage. Level 0 is where students use visualization to recognize shapes. Level 1 is where the student analyzes and compares shapes. During the 3rd level, students analyze informal arguments and begin to develop geometric truths. The last level is the highest level where students compare and contrast different axiomatic systems of geometry. These levels will give each student the skillset to advance throughout their education.

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Introducing and studying geometry allows children to use their real world observations and directly apply them into the classroom. Geometry is one topic that is prevalent in children from an early stage. Students can see, feel and navigate through shapes at an early age which van Heile mentions in the levels. These early interactions are the building blocks of a child’s knowledge with geometry. All students are surrounded by different shapes they can bring into the classroom as examples. Geometry allows students to make references for instance in measurements, “this table is 6 beach balls high” or classifying different shapes in a park with others. 

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Geometry is the language of creation and it unites everything that exists, and understanding geometric concepts makes understanding and interpreting the world around us much easier. This could be the short answer to, “Why study geometry?” Learning how to classify, sort, and analyze geometric objects and properties makes the process of describing our environment to others much easier and less time-consuming. It improves our communication and reasoning and helps us in many of our careers later in life. We also study and teach geometry because it is essential to a well-developed understanding of doing mathematics. Geometric facts and concepts make up an important interconnected facet of math that students will need to bring with them as they progress in their math learning. 
However, it is also important to extend upon this question by examining and formulating conjectures about why we study geometry (or why we teach our students geometry) in the exact manner and sequence that we do. The Van Hiele levels of geometric thought are widely used in this capacity because they enable children to continuously develop an understanding of geometric concepts that works with the natural mechanisms of their cognitive development. In other words, the Van Hiele levels, which are taught in order from most simplistic to most complex and abstract, work with students’ prior knowledge to inform each subsequent level of their geometry learning and help them to develop personal and contextual understanding of geometric objects of thought. We teach our students geometry using Van Hiele’s model because it is not only developmentally appropriate but because it reflects and aligns with the experiential construction of knowledge undertaken by every student.

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When looking at the ways in which geometry and geometric principles are accessed and used by people throughout their days, it can be hard to really quantify or comprehend how often we are utilizing certain pieces of information to make certain decisions throughout our day. Despite what field or interests a person may have, geometry is a key field within mathematics that prepares students to be able to understand the world and think critically about how two seemingly conflicting ideas can coexist at once (all squares are rectangles but all rectangles are not squares). In looking at van Hiele's levels of geometric study, it can be seen there are different foundational skills that must be established to allow for students to be able to conceptualize and internalize what the shapes are and how their properties help to defined them. By working through these levels in a sequential manner, students will be better prepared for being able to access the deductive and logical skills necessary for them in later mathematic and daily life experiences. These levels are designed to foster comprehension in students in a way that will allow them to progress through the various geometric concepts that is logical and builds upon itself.

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Why geometry?  To those 2nd graders who might not see architecture in their near future, geometry might seem abstract and unimportant.  However, through van hiele’s levels, we understand how we can not only teach geometry in a systematic and efficient way, but also one that challenges the students and fosters interest and excitement.  By going from visualization to sequential to developmental to age dependent and then finally experience dependent, students can start their geometry journey with exploration and eventually move to real world analysis.  Geometry becomes not just shapes on a paper, but a way to compare various things, how many triangles fit inside another shape.  It also pushes students to see objects with properties that can change, but that can also be finite and use this logic to make various statements.  Through van hiele’s levels, we can see geometry as something that builds on itself to become a useful addition to how students are able to address and assess the world around them.

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Geometry is critical in order to understand space, shapes and their relationships. Learning geometry helps students develop spatial sense which gives them a logical and geometric framework to describe and analyze the world. The progression of developing geometric thinking through 0. Visualization, 1. Analysis, 2. Informal Deduction, 3. Deduction and 4. Rigor provides a developmentally appropriate and stimulating thinking based learning that can be useful in students' intellectual repertoire beyond geometry. If a student gets varied experiences practicing these levels, they will become skilled thinkers.

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The study of geometry is important in all aspects because it allows you to explore and study shape and space in which we are exposed to every day, geometry also involves developing and applying spatial sense. After reading the Van Hiele levels of geometric thoughts, I realized how important and useful geometry truly is, to a point where children are expose to geometry when they are in kindergarten by starting at level 0, since they start with learning about shape and how they look like. As children go up in grades they will start to dig into the different aspects of geometry like classes of shapes, properties of shapes, relationships between properties, learning deductive systems of properties, and analysis of deductive systems. These levels are aligned with the developmental stages of a child which is why children start to learn about  geometry at a young age starting with the simpler concept of geometry until they eventually create abstract statements about geometry.       

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The study of geometry is important in all aspects because it allows you to explore and study shape and space in which we are exposed to every day, geometry also involves developing and applying spatial sense. After reading the Van Hiele levels of geometric thoughts, I realized how important and useful geometry truly is, to a point where children are expose to geometry when they are in kindergarten by starting at level 0, since they start with learning about shape and how they look like. As children go up in grades they will start to dig into the different aspects of geometry like classes of shapes, properties of shapes, relationships between properties, learning deductive systems of properties, and analysis of deductive systems. These levels are aligned with the developmental stages of a child which is why children start to learn about  geometry at a young age starting with the simpler concept of geometry until they eventually create abstract statements about geometry.       

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As students progress in their math education, calculations become less important. Students start to engage in mathematical reasoning that depends a lot on proofs and formal logic. This is especially important for types of math needed for practical fields like computer science and electrical engineering. Learning geometry is important because it provides a means by which students can progressively be introduced into these modes of thinking, especially when using the van Hiele levels.

At level zero through one, students are engaging with how the shapes look, learning about their properties, and comparing them. At this point they are beginning in very early stages of some mathematical reason. As they progress to level two, they are starting to make comparisons on the basis of those properties. Students will start constructing logical arguments in the process of these comparisons and from that they engage in early forms of proofs, even if they aren’t formal proofs. By stage three, students are introduced to the basic concepts of deductive proofs and how to engage in logical reasoning about geometry using these concepts.

Through this process geometry has provided students a context by which they can through the steps by which they can arrive at not just the concepts behind deductive reasoning/argumentation, but also why it is necessary. Geometry acts as a scaffolding for a broader higher level mathematical reasoning, that as they move on beyond high school and enter college level math they can begin to apply to a broader range of fields of math. 

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After reading Van Hiele’s characteristics of geometry his clear levels help to sort and see the importance of geometry. In level 0, students recognize shapes based on characteristics they are able to see, feel, build, or work with. In level 1 the students are able to consider the elements and characteristics of all shapes in that category rather than the singular shape in front of them. By level 2 students should be able to think about the properties of geometric objects and make comparisons between them. At level 3 students at the highschool level should be able to work with abstract statements about geometric properties and develop conclusions based on logic. Level 4 is reserved for college students majoring in geometry, these students are able to compare axiomatic systems of geometry. 
The Van Hiele levels correspond directly with the developmental stages that a child is to go through. After reading this chapter of the book it is clear to me why we teach geometry at such a young age. Students are able to grasp concepts and shape their understanding of geometry better since learning in such stages. The same way you teach numbers before addition, or teach letters before having students write sentences. Learning geometry allows students to build on prior knowledge and come to conclusions thus helping them do this in other parts of life.

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By studying geometry by
becoming grounded  in van Hiele's level of geometric thought, the student can use the levels  of visualization, analysis, informal deduction, deduction and rigor  to learn other subjects such  as linear algebra, physics and computer programming just to name a few.

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Questioning the study of geometry is valid, and can stem from an overall discomfort with mathematics. As educators we need to be prepared to justify its real-world applications. Geometry shows up in our everyday lives as tangible and three-dimensional space and shapes. The better we understand and teach geometric concepts, the more our students can see how big a role they play in their lives.

The van Hiele theory is based on how students progress in their development of geometric thinking. They established the five levels of geometric thought in order to measure and make sense of students’ types of geometric ideas. (p. 502)

The first step in the hierarchy (Level 0) is visualization. Students see and begin to group shapes. Then (Level 1) they move to the classification and analysis of shapes. Next (Level 2) comes the relationships between properties. Then (Level 3) they can develop proofs to later (Level 4) dive into axiomatic systems at the college level.

There is a developmental sequence that goes hand-in-hand with how students interpret geometric shapes and their relationships and properties. This progression can begin in kindergarten in order to lay the proper groundwork for geometry class in high school. It makes no sense to wait to teach geometrical concepts in high school when we can start early. The van Hiele levels help teachers gauge student understanding of geometry, and how we can prepare our lessons to help them get to the next level.

In the same manner that we should not teach multiplication and division without first teaching addition and subtraction, geometry should be taught in the order of levels. The kindergarteners that learn that a square is a rectangle can make better sense of geometric relationships in the higher grades.

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The Van Hiele model is a series of five levels that help students understand and conceptualize spatial ideas. The different levels help to describe the ways in which we think. Level 0 is the basic level, and students use visualization to recognize and represent items; in the case of geometry, geometric shapes. Level 1 allows students to analyze;  shapes within a class and how they are similiar/different. Level 2 allows students to make informal deductions; if a shape has 4 sides then it is a square. Level 3 allows students to deduce, or analyze arguments pertaining to shapes; they can make a conclusion that all squares are parallelograms, but not all parallelograms are squares. The final level, 4 is rigor. This is achieved at a collegiate level, and allows students to compare and contrast different properties and classifications within geometry. Studying geometry is important, as it shows the progression of Van Hiele’s levels, and allows students to build on prior knowledge. Not only that, it expands that knowledge into deeper thinking. Looking at shapes, learning about the properties of shapes, learning how to manipulate those shapes, and then being able to compare and contrast those shapes allows a student's growth. Geometry is more than just a shape. It is about abstract thinking, logical reasoning, and manipulating different shapes

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The Van Hiele model is a series of five levels that help students understand and conceptualize spatial ideas. The different levels help to describe the ways in which we think. Level 0 is the basic level, and students use visualization to recognize and represent items; in the case of geometry, geometric shapes. Level 1 allows students to analyze;  shapes within a class and how they are similiar/different. Level 2 allows students to make informal deductions; if a shape has 4 sides then it is a square. Level 3 allows students to deduce, or analyze arguments pertaining to shapes; they can make a conclusion that all squares are parallelograms, but not all parallelograms are squares. The final level, 4 is rigor. This is achieved at a collegiate level, and allows students to compare and contrast different properties and classifications within geometry. Studying geometry is important, as it shows the progression of Van Hiele’s levels, and allows students to build on prior knowledge. Not only that, it expands that knowledge into deeper thinking. Looking at shapes, learning about the properties of shapes, learning how to manipulate those shapes, and then being able to compare and contrast those shapes allows a student's growth. Geometry is more than just a shape. It is about abstract thinking, logical reasoning, and manipulating different shapes

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The study of Euclidean geometry is an excellent tool for scaffolding other forms of mathematical thinking. The stages of thinking described in the van Hiele levels correlate to the development of sensorimotor and abstract thinking in children as they age. At their youngest, students grasp the spatial and visual qualities of shapes. Because shapes also express number and operation on number, they act as a intuitive visual model of these concepts. As abstract reasoning buds in the mind, students are able to observe and articulate qualities of shapes: Triangles have three sides, squares have four. These analytical observations become more nuanced and lead to logical deductions about shapes and their classifications: all sides of a triangle intersect, so it is not a parallelogram; opposite sides of a square do not intersect, so it is a parallelogram.