Presentatie titel Geometry in the Junior Primary Okahandja, June 2016 Rotterdam, 00 januari 2007

Geometry is everywhere ……….

Everywhere is geometry……….

Geometry in the house 120o 30o 30o A ceiling fan has its 3 blades at 120o angles to make 360o while in motion. 30o 30o A Simple cloth hanger has 2 x 30o angles + 1 x 120o angle = 180o angle.

Geometry in architecture

Geometry in nature

Geometry in music

Geometry in handcraft

Geometry in cycles . Racing bikes are made using best geometry to give maximum efficiency. 

Geometry in sports

Geometry in stairs Inclined angle: 60o angle, Each stair: 90o angle.

Geometry in yoga

Geometry in tiling

Geometry in school

Geometry in the school. Which geometry do we want to teach? Grade 1 Geometrical figures Learners should be able to identify and name common 2D shapes: square, circle, rectangle and triangle, sort and classify common shapes, draw/make shapes, create pictures and patterns with shapes,

identify and name common 3D figures: sphere, cube and cone, sort and classify common 3D objects, describe common objects, create own models, find structures in the environment.   Location, position and movement of objects Learners should be able to describe the relative location of objects and people using positional and directional language, e.g. over, under, above, below, in front, behind, outside, inside, next to, between,.... [Junior Primary syllabus, Okahandja 2015]

Teaching geometry, how does it start? Van Hiele Levels of geometric thought: Level 0: Visualisation Students recognize and name figures based on the global visual charasteristics of the shape. It is the appearance of a shape that defines it for a student. ‘A square is a square because it looks like a square.’ Pierre M van Hiele [1909 – 2010] Level 1 (Visualization): Students recognize figures by appearance alone, often by comparing them to a known prototype. The properties of a figure are not perceived. At this level, students make decisions based on perception, not reasoning. Level 2 (Analysis):Level 2 (Analysis): Students see figures as collections of properties. They can recognize and name properties of geometric figures, but they do not see relationships between these properties. When describing an object, a student operating at this level might list all the properties the student knows, but not discern which properties are necessary and which are sufficient to describe the object. Level 3 (Abstraction): Students perceive relationships between properties and between figures. At this level, students can create meaningful definitions and give informal arguments to justify their reasoning. Logical implications and class inclusions, such as squares being a type of rectangle, are understood. The role and significance of formal deduction, however, is not understood. Level 4 (Deduction): Students can construct proofs Level 4 (Deduction): Students can construct proofs, understand the role of axioms and definitions, and know the meaning of necessary and sufficient conditions. At this level, students should be able to construct proofs such as those typically found in a high school geometry class. Level 5 (Rigor): Students at this level understand the formal aspects of deduction, such as establishing and comparing mathematical systems. Students at this level can understand the use of indirect proof and proof by contrapositive, and can understand non-Euclidean systems.

The red car

A square?

‘Identify and name common 2D shapes: square, circle, rectangle and triangle’ [Syllabus, grade 1] A square?

A square? Red, yellow, blue, … Wood, wool, … Small, large, … Thin, thick, … ‘A square is a square, because it looks like a square’. ‘How can you define a thing before you know what you have to define? Most definitions are not preconceived but the finishing touch of the organizing activity. The child should not be deprived of this privilege……’ Hans Freudenthal [1905 – 1990] A window, A mirror, A clock, A biscuit, A …..

A square? Not a square?

A square?

The language of geometry Talking about squares has its own vocabulary: Red, yellow, blue, purple, wood, wool, small, large, thin, thick, house, window, mirror, clock, biscuit, ….. Side, vertex, angle, plane, shape, four, opposite, equal, lenght, right angle, point of intersection, diagonals, ….. Learning geometry is a matter of language, so it should be interactive learning.

Geometry is an activity?! Learning objectives in the syllabus: Identify and name, Sort and classify, Draw/make, Create, Describe, Find. The learning objectives geometry are all skills described with verbs, so, learning geometry should be ‘learning by doing’. John Dewey [1859 – 1952] However, in modern times Dewey identified learning by doing as the sole strategy for any kind of learning. He identified education as a kind of experience, and through saying this he actually means the same equation between what one actually does and the resultant learning. So if a student learns how to listen and take notes from a lecture, he or she only learns how to listen and taking notes. If afterwards the student memorizes the lesson notes, the student learns how to memorize things. Dewey introduced his problem solving method as a strategy to teach pragmatic approach towards life problems. In problem solving method , a student through adopting a procedure solves problems . Thus, according to the equation between what one does and so one one learns, one learns how to solve problems.

Geometry in the school [2]. Which geometry do we want to teach? Grade 2 Geometrical figures Learners should be able to identify and name common 2D shapes: square, circle, rectangle, equilateral triangle and oval, sort and classify a variety of shapes according to geometrical properties: number of sides, corners, draw/make shapes, create pictures and patterns with shapes, Plaatje invoegen ‘patterns with shapes’

create recognisable 3D figures, find structures in the environment. identify and name common 3D figures: sphere, cube, cone, cylinder and rectangular prism (or brick), sort and classify common 3D objects according to geometric properties: number of faces; shape of faces, describe common objects using words such as edges, corners, faces, straight, curved, flat, round, rectangular, triangular and circular create recognisable 3D figures, find structures in the environment. Location, position and movement of objects Learners should be able to  describe the position of objects drawn on informal maps, e.g.: on a map of the classroom, my desk is on the left. [Junior Primary syllabus, Okahandja 2015] Wat kan je bouwen met 100 bekertjes? 100 cups? Practicum opdracht bouwen met cups! Zo hoog mogelijk, stevig….

Teaching geometry, how does it go on? Van Hiele Levels of geometric thought: Level 1: Analysis Students are able to consider all shapes within a class rather than a single shape. By focusing on a class of shapes, students are able to think about what makes a rectangle a rectangle. Irrelevant features fall in the background. Students begin to appreciate that a collection of shapes goes together because of its properties. Pierre M van Hiele [1909 – 2010]

Geometry in the school [3]. Which geometry do we want to teach? Grade 3 Geometrical figures Learners should be able to identify and name common 2D shapes: square, circle, rectangle, equilateral triangle and oval and pentagon, sort and classify a variety of shapes according to geometrical properties: number of sides, corners, angles, right angles, Identify right angles by use of a reference tool, draw/make shapes, create pictures and patterns with shapes, identify and name 2D shapes: square, circle rectangle, triangle, equilateral triangle, oval and pentagon sort and classify a variety of shapes (quadrilaterals, pentagons, hexagons, octagons) according to geometrical properties: number of sides, side lengths, number of vertices, number of angles, number of right angles (learners do not need to know the terms ‘hexagon’ and ‘octagon’) identify right angles; use a reference tool (paper corner, pattern block) to identify angles in shapes as greater than, equal to or smaller than a right angle draw/make shapes showing essential spatial features, e.g. join five sticks/straws of equal length to form a recognisable pentagon

identify and name common 3D figures: sphere, cube, cone, cylinder and rectangular prism (or brick), pyramid, sort and classify common 3D objects according to geome- tric properties: number of faces, edges and vertices, describe common objects using words such as edges, corners, faces, straight, curved, flat, round, rectangular, triangular and circular, make models from paper and recycled material, find structures in the environment, recognise angles in everyday objects. Location, position and movement of objects Learners should be able to  describe the relative location of objects drawn on maps of familiar settings. [Junior Primary syllabus, Okahandja 2015]

Teaching geometry, how does it go on? Van Hiele Levels of geometric thought: Level 2: Analysis Students perceive relationships between properties and between figures. Students can create meaningful definitions and give informal arguments to justify their reasoning. Logical implications and class inclusions, such as ‘squares being a type of rectangle’ are understood. Pierre M van Hiele [1909 – 2010] Level 3 (Abstraction): Students perceive relationships between properties and between figures. At this level, students can create meaningful definitions and give informal arguments to justify their reasoning. Logical implications and class inclusions, such as squares being a type of rectangle, are understood. The role and significance of formal deduction, however, is not understood. Level 4 (Deduction): Students can construct proofs Level 4 (Deduction): Students can construct proofs, understand the role of axioms and definitions, and know the meaning of necessary and sufficient conditions. At this level, students should be able to construct proofs such as those typically found in a high school geometry class.

Implications for Geometry education Geometry education should be interactive to use the language of geometry, Learning by doing to experience and to master the ‘geometric skills’, Pleasure in solving problems, reasoning, discovering properties, seeing the aesthetic aspects of geometry, Discover the magnificient presence of geometry in nature and daily life. So the teacher should organise a lot of interactive hands-on activities for his learners and discuss the experiences by ‘doing geometry’.

Summary: Subjects to teach in the Junior primary Shapes 2-D shapes, 3-D shapes Location

How to teach ‘Shapes’? Discover shapes in everyday life, Construct models, Design patterns, Design a tiled floor, Play games, Fill out worksheets, Make geometric puzzles, Read a math storybook, ……….

How to teach ‘Location’? Create a tableau, Dance symmetric, Play ‘hide-and-seek’, Walk a route, Look at the sun and the shadows, Look at the mirror, Fill out worksheets, Read a math storybook, ……….

Geometry education – Learning by doing

We get to work with geometry!