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Students often have poor achievements in mathematics due to an improper sequence of instructions and an ineffective choice of teaching methods. This paper analyzes Van de Walles sequence of instruction, Van Hieles theory of geometric thought and Steven Leinwands instructional shifts. Apart from determining the effectiveness of these approaches in the classroom, specific examples of their use are presented. Only a combination of all three methods can help to achieve positive outcomes.

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The Use of Van de Walles Sequence of Instruction in the Classroom

According to Van de Walles sequence of instruction, it is efficient to use three steps for measuring instruction. The first step involves making comparison. The objective of this step is to help students understand how the attributes of particular things can be assessed. For example, students learn to determine, which objects are longer/shorter. This activity allows them realize the concept of length. The first step of instruction corresponds to the Common Core State Standards because it makes learning of the material clear and understandable for students. In fact, the first step prepares students for learning new concepts. The second step of the Walles instruction applies some models of measuring standard units. At this stage students start to realize how to make measurements by using numbers. Finally, the third step enables to apply measuring tools. The objective of this step is to teach students to use typical measuring instruments easily. For example, students learn to use such standard tools as rulers, scales, and protectors.

As far as my own measurement instruction is concerned, it primary corresponds to the Common Core State Standards, due to the fact that it is clear and coherent. I try to apply the findings of researches in instruction and consider requirements of colleges and employers while teaching measurement. For example, I follow Walles sequence of instruction because studies prove that it offers a gradual measurement instruction and it is suitable for children with different level of skills (Hine et al, 2016, p.174). Besides, Walles approach to instruction also helps me to develop the problem-solving skills of children and prepare them for life after leaving a school (Flores, Hinton, & Burton, 2016, p.245). In addition, I use instructional shifts of Steven Leinwand, because they help to follow the standard of allowing students to utilize the learnt information through thinking skills of a higher order.

Teaching volume is the example of incorporating Walles sequence of instruction in my own classroom. The first step involves making direct comparisons of few objects. In particular students compare the volume of two boxes by putting one box inside another one. If the class is not performing well, we revise the process of making comparisons of perimeter and area prior to proceeding to volume. The next step is determining the volume of an object by using nonstandard measurement units (e.g. sugar cubes). Then students practice using standard units of estimating volume. For instance, they apply centimeter cubes to determine the volume of an object. In this case, the volume of each cube is 1 cm3. All these steps help students to understand what volume is and how to measure it by using standard and nonstandard units and instruments.

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Van Hieles Levels of Geometric Thought for Teaching Geometry Concepts

Van Hieles theory of geometric thought presents five levels of knowledge, which students gain during studying geometric concepts. The zero level is called visualization/recognition. Students learn to describe the shape of some objects, depending on the way they look. The first level is analysis/descriptive. Students learn to describe the shape, considering the specific characteristics of an object. The second level is informational deduction or abstraction. At this level, children identify the importance of characteristics and relations between them. It allows them putting the characteristics of shapes in an order. The third level is deduction. At this level of geometric thought, students gain logical reasoning ability and become able to examine theorems deductively (Howse & Howse, 2014, p.310). The fourth level is rigor. It is a level at which students can set and examine theorems in various postulate systems. Similarly to the Van de Walles sequence of instruction, Van Hieles levels of geometric thought offer moving from simple degrees of understanding to more difficult ones. Therefore, this approach is useful for teaching new geometry concepts.

Van Hieles levels of geometric thought could be also used in my own classroom. During the first phase of teaching geometric concepts, it would be reasonable to ask about geometric shapes, which are familiar to students. Then they can participate in few sorting exercises. For example, children may be offered to determine the properties of the blocks (e.g. color, shape, size). Thus, some students would sort blocks by their size and color. Other students would sort blocks with or without corners. At the end of the phase, students will have to discuss all the properties of each object.

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The example of the activity for the second level is searching objects with one, two or three attribute differences. In order to make the activity more interesting, it is reasonable to do the exercise in the form of a game. For instance, one student can demonstrate one small, thick, red circle and one small, thin, red triangle to others. The rest of the group would have to guess attribute differences, which are shape and thickness in this case.

At the next stage, students should work in small groups. Each group will have to put all small blue objects in the intersection between the small and the blue circles of a Venn diagram. Such activity allows children to practice in verbalizing their understanding of geometric concepts, because children are encouraged to discuss their decisions. The list of questions: Why do these objects fit there? Are there any objects that do not fit there? Why?

The next level involves asking students to make their own Venn diagram situations. The examples of diagrams created by students are red and big, thin and blue, thick and circle. During the drawing process, a teacher can move around the classroom asking some questions. The list of questions: What kind of shapes would be present in the intersecting circle if one circle presents red shapes and another circle presents blue shapes? How do shapes in the intersecting circles differ from the shapes in other circles?

At the last level, students can play a game, which resembles Scrabble. However, children have to use a Venn diagram with two or three intersecting circles instead of a board game with letters. As well as in Scrabble children take attribute blocks from a bag. The goal of the game is to have the smallest number of blocks in the end. Students place a block in certain circle of the diagram and explain their choice to others.

The Use of Instructional Shifts of Steven Leinwand

Steven Leinwand offers ten instructional shifts, but the most helpful techniques for my classroom are conducting daily cumulative reviews and moving beyond one correct response in order to encourage students thinking as well as provide opportunities for drawing, modeling, and visualizing math concepts. I have chosen the first instructional shift because my students tend to forget the learnt material fast, so regular cumulative reviews will help them to remember all the important information (Leinwand, 2009, p. 5). For example, I will incorporate some mini-quizzes and follow-up discussions. In order to revise geometry concepts, I will demonstrate different objects and ask students about their attributes.

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I have chosen the second instructional shift, because it encourages students to think, and it is important for memorizing the material and improving its further use in practice. This approach is effective for checking homework assignment. For example, I can ask students: How did you get that? Does everybody agree? Does anybody have another answer? What do you think…? However, it is necessary to select specific tasks for homework, because it is not possible to create questions for all assignments. For instance, I will offer my students to create their own Venn diagram situations at home and then use them in tasks in the classroom. During fulfilling the task I will move around the class and ask students to explain their decisions.

I have chosen the third instructional shift because it raises the interest of students in Mathematics due to the fact that most children like drawing (Lehmann, 2015, p.78). This instructional shift could be used for teaching various concepts. For example, students can be asked to draw an apple and then point 1?2, 1 and 1?2 or 3?4 of it. Besides, I will apply this instructional shift while explaining shapes and their properties.

In conclusion, the usage of Van de Walles sequence of instruction, Van Hieles theory of geometric thought, and Steven Leinwands instructional shifts can significantly enhance mathematics achievement in my classroom. Thus, Van de Walles approach can help me to present new material, because it allows moving gradually from simple activities to the difficult ones. Van Hieles theory of geometric thought is also helpful for efficient teaching of some new geometric concepts. However, this approach requires much time, because it includes five levels. The instructional shifts of Steven Leinwand have different functions. There are some recommendations that can improve the teaching process. In particular, Leinwand advises to make some cumulative reviews daily in order to prevent students from forgetting the material of the previous lessons. Another idea is asking some questions, which have one right answer. A teacher can use this approach for checking home assignments. Finally, it is effective to draw, model, and visualize the concepts. Not only does this method improve students understanding but also raises the interest of students in a lesson.