The Problem• Students in classes enter at varying levels of understanding

• Textbooks often introduce concepts in ways that are confusing to students

• Students do not understand the purpose of axioms and definitions• Students can often do problems without understanding the concepts behind them

• The SOLs expect students to be at a level 3 by the end of sixth grade. However, most text books only go until level 1

Who was Van Hiele?• Dina van Hiele-Geldof and her husband Pierre Marie van Hiele

• Dutch educators

• Were frustrated by making no progress with current methods of teaching

• Reorganized the way algebra was taught and had a lot of success

• i.e. x4 * x2 = x6 but (x4)2

= x8

• Tried to use similar methods for geometry, but found little success

Problems with the Current Theories

• The psychology of Piaget was one of development and not of learning. So the problem of how to stimulate children to go from one level to the next was not his problem

Problems with the Current Theories

• Piaget distinguished only two levels. In geometry it appears necessary to distinguish more. Some of Piaget’s results would have been more intelligible if he had distinguished more than two levels.

Problems with the Current Theories

• Paget did not see the very important role of language in moving from one level to the next. It was occasionally suggested to him that children did not understand his questions. He always answered that they did understand; this could be read from their actions. But, although actions might be adequate, you cannot read from them the level at which children think

Problems with the Current Theories

• According to Piaget, human spirit develops in the direction of certain theoretical concepts. He was not aware that those concepts are only human constructions, which, in the course of time, may change. So development with some theory as a result always must be understood as a learning process influenced by people of the period

Problems with the Current Theories

• Piaget did not see structures of a higher level as the result of study of the lower level. In the van Heile theory, the higher level is attained if the rules governing the lower structure have been made explicit and studied, thereby themselves becoming a new structure. In Piaget’s theory, the higher structure is primary; children are born with it, and only have to become aware of it.

Problems with the Current Theories

• In the van Hiele Theory, a structure is a given thing obeying certain laws (borrowed from Gestalt theory); if it is a strong structure it will usually be possible to superpose a mathematical structure onto it. In Piaget’s theory the mathematical structure always defines the whole structure.

Van Hiele Theory• Proposed that students go through 5 levels in learning geometry

• The learner can not achieve one level without passing through the previous levels

• Progress from one level to another is more dependent on educational experience than on age or maturation

• Certain types of experiences can facilitate or impede progress within a level or to a higher level

Van Hiele Levels

Level 0?

• Clements and Battista Proposed the existence of a Level 0 in 1992

• Pre-recognition

• Although not an official part of the original van Hiele model, it has gained wide acceptance and is usually refered to as part of the model.

Level 0

• Pre-recognition – Students at this level notice only a subset of the visual characteristics of a shape, resulting in an inability to distinguish between figures. For example, they may distinguish between triangles and quadrilaterals, but may not be able to distinguish between a rhombus and a parallelogram

• Shapes fit in pre-existing stereotypes gained from earlier childhood experience.

• Example:

Level 1

• Visualization – Geometric figures are recognized as entities, without any awareness of parts of figures or relationships between components of the figure. A student should recognize and name figures and distinguish a given figure from others that look somewhat the same.

• “I know it’s a rectangle because it looks like a door and I know that the door is a rectangle”

Level 2

• Analysis – Properties are perceived, but are isolated and unrelated. A student should recognize and name properties of geometric figures.

• “I know it’s a rectangle because it is closed, it has 4 sides and 4 right angles, opposite sides are parallel, opposite sides are congruent, diagonals bisect each other, adjacent sides are perpendicular, …”

Level 3

• Abstraction – Definitions are meaningful, with relationships being perceived between properties and between figures. Logical implications and class inclusions are understood, but the role and significance of deduction is not understood.

• “I know it is a rectangle because it’s a parallelogram with right angles”

Level 4

• Deduction – The student can construct proofs, understand the role of axioms and definitions, and know the meaning of necessary and sufficient conditions. A student should be able to supply reasons for steps in a proof.

Level 5

• Rigor – The standards of rigor and abstraction represented by modern geometries characterize level 5. Symbols without referents can be manipulated according to the laws of formal logic. A student should understand the role and necessity of indirect proof and proof by contrapositive.

Properties of the Levels• Adjacency – what was intrinsic in the preceding level is extrinsic in the current level

• Distinction – each level has its own linguistic symbols and its own network of relationships connecting those symbols

• Separation – Two individuals reasoning at different levels can not understand one another

• Attainment – the learning process leading to complete understanding at the next higher level has five phases: inquiry, directed orientation, explanation, free orientation and integration

The Five Phases of Attainment1) Information (Inquiry) – Gets acquainted with the working domain (e.g., examines examples and non-examples

2) Guided (directed) Orientation – Does tasks involving different relations of the network that is to be formed (e.g., folding, measuring, looking for symmetry)

3) Explication (Explanation) – 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)

4) Free Orientation – 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 kites)

5) Integration – Summarizes all that has been learned about the subject, then reflects on actions and obtains an overview of the newly formed network of relations now available (e.g., properties of a figure are summarized)

Usiskin (1982)

• In 1982, Usiskin examined 2,699 students whom were enrolled in 99 high school geometry classes. The Poll included 13 different schools over 5 different states.

• He was able to assign a van Hiele Level to 88% of these children a CDASSGP test

• Some students were transitioning between levels and were difficult to classify

• 40% of students finishing high school geometry were below level 3.

Mayberry (1983)

• Polled 19 preservice elementary teachers and concluded that the evidence supports the hierarchical aspect of the theory

• Rejected the hypothesis that an individual demonstrated the same level of thinking in all areas of geometry included in school programs

Table 6% of Students Identifying Various Shapes as Squares________________________________________________ Shape _____________________________________Grade n

________________________________________________8 20 100.0 100.0 20.0 0.0 0.0 0.07 24 100.0 100.0 8.3 4.2 4.2 0.06 20 100.0 100.0 30.0 10.0 10.0 0.0________________________________________________Total 64 100.0 100.0 18.8 4.7 4.7 0.0________________________________________________

Table 8% of Students Identifying Various Shapes as Rectangles____________________________________________ Shape ____________________________________________Grade n____________________________________________8 20 40.0 40.0 10.0 100.0 100.0 45.07 24 50.0 50.0 0.0 100.0 100.0 4.26 20 55.0 55.0 10.0 95.0 95.0 40.0___________________________________________________Total 64 48.4 48.4 3.1 98.4 98.4 28.1

____________________________________________

Number of Students At Each Grade Level Providing Specific Definitions of “Isosceles Triangle”________________________________________________________________

Grade

________________________6 7 8 Total

________________________________________________________________At least 2 sides = 5 8 3 162 sides =, interpreted as “at least 2” 2 7 7 162 sides =, interpreted as “exactly 2” 3 1 4 82 sides =, inconsistent interpretation 2 2 0 42 sides = with 3rd different 0 1 4 5No sides congruent 2 2 2 6All sides congruent 2 0 0 22 equal angles 2 8 4 14All angles < 90° 2 0 0 2One angle > 90° 0 2 0 2I don’t know 1 0 1 2________________________________________________________________

Table 1% of Subjects at Each van Hiele Levelas Determined by the CDASSGP Test_______________________________________________

van Hiele Level ______________________________________________

notmastered

Grade n 1 1 2 3 4 5 no-fit_______________________________________________8 46 0.0 15.2 10.9 26.1 2.2 10.9 34.87 36 2.8 8.3 8.3 25.0 5.6 2.8 47.26 38 7.9 23.7 23.7 7.9 7.9 2.6 26.3_______________________________________________Total 120 3.3 15.8 14.2 20.0 5.0 5.8 35.8

Table 2% of Gifted Students and Students EnteringHigh School Geometry at Each van Hiele Levelon the CDASSGP Test Excluding “No-Fits”______________________________________________

van Hiele Levelnotmastered

Grade n 1 1 2 3 4 5______________________________________________

8 30 0 23 17 40 3 177 19 5 16 16 47 11 56 28 11 32 32 11 11 4

______________________________________________Total 77 5 25 22 31 8 9______________________________________________ HighSchool* 241 27 51 15 7 0 0______________________________________________* Data for students entering high school geometry was reported by Senk (1989, p. 315). Other data is from gifted students in the current study.

Solution

• Gizmos - http://www.explorelearning.com

• Geometers Sketchpad

• Technology – Why?

• Interviews

• Open ended word problems

Bibliography

Crowley, Mary L. The van Hiele Model of the Development of Geometric Thought. (1987). Retrieved January 30, 2008, from http://www.explorelearning.com

Van Hiele, Pierre. Structure and Insight: A Theory of Mathematics Education. Academic Press, Inc. New York: 1986.