# Geometry and Spatial Sense

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1 Geometry and Spatial Sense Diagnostic Assessment Grade Level: Appropriate for grades 4-6 Assessing the Students Levels of Thinking Based on the van Hiele Model

2 The Rationale The Van Hiele Theory According to the van Hiele theory (1986), with appropriate instructions, there are five distinct levels of understanding through which students progress in relation to geometry (Crowley, 1987). A student could demonstrate geometric thinking that fits into any of the Van Hiele levels, at any age. The development of geometric thought based on this model, was created in the 1980s by two Dutch middle school teachers and researchers, Dina van Hiele- Geldhof and Pierre Van Hiele. The Van Hiele levels begin with the most basic component visualization which is the ability to see geometric shapes as a whole without focusing on their particular attributes, and continue to the most advanced level rigor, which is the ability to learn that geometry needs to be understood in the abstract (Van Hiele 1999). This model is unrelated to the children s developmental skills or stages of thinking where students must reach a certain age to progress through the levels; rather, it is related to the individuals prior experiences through activities and opportunities (Van Hiele 1999). In other words, students graduate from their current levels to the next one through practice rather than age, and it is crucial for teachers to provide experiences and tasks that allow students to develop accordingly. It is the teacher s responsibility to maintain awareness of each individual student s Van Hiele level of thinking through proper observation and record keeping. The teacher can then design geometry tasks to match the student s level and eventually raise the child s thinking to a higher level. How to Apply the Theory Through the proper implementation of the Van Hiele theory, the teacher will be provided with an opportunity to familiarize his/herself with the students strength and areas of improvement. The teacher will then be able to facilitate a student s achievements toward further progress by asking questions at the next higher level. According to Mason (1997), using lecture and memorization as the main methods of instruction will not lead to effective learning.

3 Teachers should provide their students with appropriate experiences and the opportunities to discuss them (such as using manipulatives or group work). In addition, as explained by Mayberry in The van Hiele Levels of Thought in Undergraduate Preservice Teachers, The teacher should provide experiences based on the phases of learning to develop each successive level of understanding because if a teacher teaches a level of thought that is above the students understanding, the student will fail to comprehend the content. If a student is being taught a content that is above their level of comprehension, it is likely that he/she will try to memorize the material without truly mastering the content. Students may easily forget material that has been memorized, or be unable to apply it, especially in an unfamiliar situation. When teaching mathematical concepts, teachers must maintain careful consideration of the students progress and apply differentiated instructions depending on the skills of the students within the class in order to lead them through the necessary experiences to move on. The Assessment According to the Van Hiele model, because the phases of learning are recurring, they do not necessarily correspond to certain types of assessment thus teachers need to strive to make assessment an ongoing, daily aspect of the teaching process, rather than an occasional interruption. The following assessment is one which could be administered as a diagnostic, formative or summative to determine the students geometric level of thinking. This assessment is appropriate for the junior grades as the questions only range from levels 0 to 2 of the Van Hiele s model of thinking. Furthermore, this assessment is one on one rather than pencil paper which creates the opportunity for the use of manipulatives, as well as the assessment of the students thought process. For instance, the second question at station one requires the students to distinguish between the symmetrical and non symmetrical shapes, which fits into the level 0 visualization category, which is the most basic level and means the student can distinguish a shape by its appearance. For students who successfully give the correct response to the question and show that they have mastered the level 0 thinking model of Van Hiele, they will then be prodded forward by being asked level one questions such as How do you know shape N is not symmetrical?, encouraging the use of descriptive thinking.

4 Asking the student to explain how they knew a certain mathematical concept, could pave their way to make certain logical claims. While some students may describe the properties of shapes using level 1 thinking, gifted children might be pressed to classify a set of shapes on the basis of these properties or to deduce ideas and relationships about the shapes (Geddes and Fortunato, 1993). It is reasonable to assume that such children are likely to develop their own definitions of certain shapes or to deduce relationships between some shapes such as a triangle and a quadrilateral. In conclusion, such assessments are needed to provide insight into how students understand definitions and how such definitions are used in making and supporting their claims. The following assessment and the questions that the students will be asked at the two stations, support the van Hiele model of student geometric thinking.

9 Station Two (Interview Sheet) Big Idea At this station the students will be assessed on their knowledge and comprehension of Location and Movement. The students will be asked to identify shapes in various transformations such as a reflection, rotation, turn, flip or slide. Materials Student Worksheet, Pencil, Tanagram pieces, Geoboard, Elastics 1. Show Tanagram pieces and Ask the Student to change the initial position of the geometric shape, by either turning, flipping or sliding the shape. Say: Can you show me a turn, flip and slide of this shape? Extension: Ask the student how knew how to make a turn, flip and slide. Ask: Does a turn, flip and slide remain you of another type of transformation and why? Correct Answer: A Turn can vary A Flip should look like Slide would look like Correct Answer: If the student does not show the correct transformation but could not explain how they knew, the student will receive Level 0, do not ask further questions. If the student can describe that a turn requires the shape to stay in the same position and move from left to right, they would receive Level 1. If the student could compare a turn with a rotation or a flip with a reflection that would display higher thinking and the student would receive Level 2.

10 2. Using a Geo board, the student will have a chance to demonstrate their understanding of turns. Ask the student to make a turn of the pentagon below using a Geo board. Say: Can you rotate this shape on the Geo board using the elastics? Correct (Answer) is a Rotation 90 degree rotation If the student does not create the correct rotation move on to the next question, if the student could not explain how they knew how to make a Rotation the student would receive Level 0. If the student was able to rotate the shape and explain how they knew what to do, score the student with a Level Show the students a picture of a shape in the initial position and then in its moved position. Ask the students to identify what movement the following shape is in. Say: The Triangle in the Initial Position has moved. What type of movement is it in now? A Initial Position Has the triangle moved into a B C a) Slide b) Turn B C c) Reflection A d) Flip Correct Answer: Answer: Reflection. If the student could not explain why they chose the Reflection the student will receive Level 0. If the student could explain that the shape is a reflection because it crossed the grid and every point is in the same position except A then the student will receive Level 1.

13 Student Worksheet for Station One 1. The students should be assessed on their ability to identify various geometric shapes and their properties. Ask: What are the names of these shapes? Show: Show the students various shapes and provide students with the names of the shapes. Ask: Can you match the name of the geometric shape with its correct shape? Words Hexagon Trapezoid Right Angled Triangle Parallelogram 2. Can you place a chip on the shapes that are symmetrical? 3. Can you draw the line of symmetry for each of these shapes? Remember there are can more than one line of symmetry for each shape or there could be no lines of symmetry at all. 4. You get to create your own symmetrical shape! Can you draw half of the shape on the left side of the line and its symmetrical half on the right? Remember, it has to be a symmetrical shape.

14 5. Can you show me which shape is congruent with Figure A? Remember, there could be more than one answer here. Figure A Shapes 6. Can you identify the congruent angles on each right angled triangle? 7. How many sides does this shape have? How many angles? How many vertices?

15 Student Worksheet at Station Two 1. Can you show me a turn, flip and slide of this shape? Correct Answer: 2. Can you make a rotation of the pentagon on the Geo board using the elastics? 3. The Triangle in the Initial Position has moved. What type of movement is it in now? A Initial Position Has the triangle moved into a B C a) Slide b) Turn B C c) Reflection A d) Flip

16 4. Can you draw this shape again in a 180 degree angle to the right? 5. Can you draw a translation for the triangle on the grid? What did you have to remember when drawing the translation? Could you draw the translation 3 unit up?

17 Scoring Chart for Van Hiele s Geometric Levels of Thinking Question Level 0 Level 1 Level 2 Notes/Comments Station One Station Two

18 The van Hiele model of geometric understanding

19 Self Directed Learning Assessment Rubrics Rationale [ worth 10 marks] Quality of Writing [Worth 5 marks] < % % % % Mark The rationale is The rationale The rationale The rationale The rationale missing includes includes some includes includes a relevant and unclear relevant sufficient variety of detailed information information relevant and relevant and information with vague but is missing detailed detailed and as well as details. The important information information citations citations are details and with citations with proper minimal and correct citations incorrect citations The assignment is extremely unclear with no clarity in the presentation of ideas The assignment is disorganized and lacks cohesion in ideas The assignment lacks clear presentation of ideas The assignment is well written with considerable degree of clarity The assignment is articulate and written with high degree of clarity The assignment Is filled with spelling and grammatical errors The assignment has several spelling and grammatical errors The assignment has some spelling and grammatical errors The assignment has a few spelling and grammatical errors The assignment is free of spelling and grammatical errors Organization of research sources [Worth 5 marks] References are not listed References are disorganized and difficult to navigate References are show some organization References are organized and easy to access References are very organized and easy to access Assessment Script [Worth 10 marks] There is no evidence of the Van Hiele theory in the script There is limited evidence of the Van Hiele theory in the script There is some evidence of the Van Hiele theory in the script There is evidence of the Van Hiele theory in the script There is clear evidence of the Van Hiele theory in the script The script is disorganized, vague and impossible to follow The script is poorly organized, with minimal details, and is difficult to follow The script is somewhat organized with a few details, and complicated to follow The script is considerably organized, sufficiently detailed and easy to follow The script is highly organized, extremely detailed and very easy to follow Self- Created Assessment [Worth 5 marks] This assessment rubric is disorganized and does not include any of the necessary criteria This assessment rubric is poorly organized and includes only one or two of the necessary criteria This assessment rubric is somewhat organized, and includes some of the necessary criteria This assessment rubric is considerably organized, thorough and includes most of the necessary criteria This assessment rubric is highly organized, thorough and includes all the necessary criteria

20 Assessment Rubric for Activity Stations Assessment Questions [ Worth 10 marks ] < % % 70 79% % Mark The questions are The questions The questions are The questions are All The questions are not prepared in any follow a poor order prepared in a prepared in a prepared in a highly logical order of of geometrical somewhat logical considerably logical logical order of geometrical concepts order of order of geometrical concepts geometrical geometrical concepts concepts concepts None of the questions are reflective of the Van Hiele s model of thinking Only one or two questions are reflective of the Van Hiele s model of thinking Some of the questions are reflective of the Van Hiele s model of thinking Most of the questions are reflective of the Van Hiele s model of thinking All the questions are highly reflective of the Van Hiele s model of thinking Use of Manipulatives [Worth 8 marks ] There are no useful manipulatives incorporated in the assessment Only one useful manipulative is incorporated in the assessment A few useful manipulatives are incorporated in the assessment Some useful manipulatives are incorporated in the assessment A variety of useful manipulatives are incorporated in the assessment Modifications /Extensions [ Worth 6 marks ] The levels of difficulty for none of the questions are modified/extended based on the anticipated student responses The levels of difficulty for one or two of the questions are modified/extended based on the anticipated student responses The levels of difficulty for some of the questions are modified/extended based on the anticipated student responses The levels of difficulty for most of the questions are modified/extended based on the anticipated student responses The levels of difficulty for all the questions are modified/extended based on the anticipated student responses Student assessment sheet [ Worth 6 marks ] The assessment sheet is disorganized and not coherent with the Van Hiele model The assessment sheet for the students is poorly organized and minimally coherent with the Van Hiele model The assessment sheet for the students is somewhat organized but is minimally coeherent with the Van Hiele model The assessment sheet for the students is considerably organized and coherent with the Van Hiele model The assessment sheet for the students is highly organized and coherent with the Van Hiele model Please write your comments below. Total: / 65 Cheers: Next Steps:

21 References Crowley, Mary L. The van Hiele Model of the Development of Geometric Thought. In Learning and Teaching Geometry, K-12, edited by Mary Montgomery Lindquist, pp Reston, VA: National Council of Teachers of Mathematics, 1987 Geddes, Dorothy. Geometry in the Middle Grades. Curriculum and Evaluation Standards Addenda Series, Grades 5 8. Reston, Va.: National Council of Teachers of Mathematics, Mason, M. (1997). "The van Hiele level of Geometric Understanding". Journal for the Education of the Gifted, (21) Math is Fun. Retrieved February 17, Mayberry, J. The van Hiele Levels of Thought in Undergraduate Preservice Teachers. Journal for Research in Mathematics Education, 14: pp Ontario Ministry of Education. Geometry and Spatial Sense, Grades 4 to 6. Queens Printer for Ontario, van Hiele, Pierre M. Developing Geometric Thinking through Activities That Begin with Play. Teaching Children Mathematics 5 ( January 1999):

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