Geometry Trajectory It All Together –Putting Each concept builds on the previous idea and students should explore and construct concepts in such a sequence Level 0 *Kindergarten Level 1 Analysis Level 2 Information Level 3 Level 4 Rigor Level Deduction Deduction The object of thought is The objects of classes of shapes. thought are the Visualization The objects of thought The objects of deductive are the properties. thought are the The object of thought Students consider all axiomatic – relationships shapes in a class instead of systems. Students understand among the (focus) is individual shapes. – an individual example. that properties are properties. The products of Students are operating on – related. What makes a shape – “a thought are specific examples of the The products of shape”? comparisons and One set of properties shapes they see. – thought are the WALLE WALLE may relate to another contrasts among deductive The outcome is discovering DE DE Orientation may even – different property. axiomatic properties of shapes. change what a student axiomatic systems systems. VAN VAN Students can observes. – of geometry. Focus on properties. – understand logical The goal is to see – Apply ideas to entire reasoning about the – likenesses and differences. classes of figures. properties. The outcomes are relationships among properties. Adapted from the and taken from van de Walle & Lovin van Hiele Levels of Geometric Thought Teaching Student-entered Mathematics (k-3) The Big Picture in Kindergarten
Geometry Trajectory –Putting It All Together Each concept builds on the previous idea and students should explore and construct concepts in such a sequence VAN DE WALLE VAN DE WALLE Level 0 *Kindergarten Level Visualization The object of thought (focus) is individual shapes. – Students are operating on specific examples of the shapes they see. – Orientation may even change what a student observes. – The goal is to see likenesses and differences. Level 1 Analysis The object of thought is classes of shapes. – Students consider all shapes in a class instead of an individual example. – What makes a shape “a shape”? The outcome is discovering properties of shapes. – Focus on properties. – Apply ideas to entire classes of figures. Level 2 Information Deduction The objects of thought are the properties. – Students understand that properties are related. – One set of properties may relate to another property. – Students can understand logical reasoning about the properties. The outcomes are relationships among properties. Level 3 Deduction The objects of thought are the relationships among the properties. The products of thought are the deductive axiomatic systems. Level 4 Rigor The objects of thought are the deductive axiomatic systems. The products of thought are comparisons and contrasts among different axiomatic systems of geometry. Adapted from the van Hiele Levels of Geometric Thought and taken from van de Walle & Lovin Teaching Student-entered Mathematics (k-3) The Big Picture in Kindergarten