Geometry. Professor Harms Minnesota State University Moorhead Feb. 24 th, 2014

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1 Geometry Professor Harms Minnesota State University Moorhead Feb. 24 th, 2014

2 Engaging those Calvins in your class

3 The van Hiele model of thinking in Geometry The student recognizes, names, compares and operates on geometric figures according to their appearance. The student analyzes figures in terms of their components and discovers properties/rules of shapes. The student logically interrelates properties/rules by giving informal arguments for relationships. The student proves theorems deductively and establishes a formal mathematical system. The student use an axiomatic system to work in an abstract level.

4 Progress from One Level to the Next Involves Five Phases 1. Information: The student gets acquainted with the working domain (e.g., examines examples and non-examples).

5 Progress from One Level to the Next Involves Five Phases 2. Guided orientation: The student does tasks involving different relations of the network that is to be formed (e.g., folding, measuring, looking at how the number of edges faces and vertices compare).

6 Progress from One Level to the Next Involves Five Phases 3. Explication: The student becomes conscious of the relations, tries to express them in words, and learns technical language which accompanies the subject matter (e.g., expresses ideas about properties of figures).

7 Progress from One Level to the Next Involves Five Phases 4. Free orientation: The student learns, by doing more complex tasks, to find his/her own way in the network of relations (e.g., knowing properties of one kind of shape, investigates these properties for a new shape, such as tetrahedrons).

8 Progress from One Level to the Next Involves Five Phases 5. Integration: The student summarizes all that he/she has learned about the subject, then reflects on his/her actions and obtains an overview of the newly formed network of relations new available (e.g., properties of a figure are summarized).

9 Major Characteristics of van Hiele levels: The levels are sequential Each level has its own language What is implicit at one level becomes explicit at the next level Material taught to students above their level is subject to reduction of level Progress from one level to the next is more dependent on instructional experience than on age or maturation One goes through various phases in proceeding from one level to the next.

10 MN Math Standards Benchmarks Grade 1 Benchmark: Characteristics of Shapes Describe characteristics of two- and three-dimensional objects, such as triangles, squares, rectangles, circles, rectangular prisms, cylinders, cones and spheres. Benchmark: Compose & Decompose ShapesCompose (combine) and decompose (take apart) two- and threedimensional figures such as triangles, squares, rectangles, circles, rectangular prisms and cylinders. For example: Decompose a regular hexagon into 6 equilateral triangles; build prisms by stacking layers of cubes; compose an ice cream cone by combining a cone and half of a sphere. Grade 3 Benchmark: Polygons Sketch polygons with a given number of sides or vertices (corners), such as pentagons, hexagons and octagons.

11 Levels of van Hiele Theory Recognition Analysis Relationships Deduction Axiomatic Mnemonic Red ants remove dirt all-around

12 Progression in van Hiele Information Guided orientation Explication Free orientation Integration

13 Resources Continued Getting to Know the Shapes-Lesson Plan with Handout for Junior Architects Geometric Solids and Their Properties- Lesson Plan with Handout and Java Applet Math Net Arrange cubes and see the top, front, and right perspectives of the stacked cubes A + Math has a Geometry MATHO game that can be used as a review of plane (two dimensional) geometry terms.