In a previous post I address the problems that lead many students to struggle in geometry and outlined a solution based on the van Hiele’s research. In this article I will summarize their findings and suggest ways to implement instruction that will
move students to acquire higher levels of geometric thinking.
The van Hiele’s noted that students climb through the five levels of geometric knowledge outlined here.
1. Visualization – Children can identify shapes based on appearance, not on properties. Students at this level may not see a square as a type of rectangle nor even recognize it as a square if it is rotated slightly.
2. Analysis – At this level, students begin to associate properties with their shapes. The student who struggled to identify a rotated square will now see that it has four congruent sides and four right angles and is therefore a square.
3. Abstraction – Now students can begin to think about the properties and apply them to arguments that involve inductive reasoning. The student who sees that four different triangles all have an interior angle sum of 180° would use that pattern to reason that all triangles must have the same interior angle sum.
4. Deduction – At this level, students use deductive logic to prove their conjectures from the previous level.
5. Rigor – This goes beyond the former level to explore proofs by negation and non-Euclidean geometry.
Activities for level one:
· Provide students with physical examples of two- and three-dimensional shapes.
· Expose students to a wide variety of geometric shapes in both two and three dimensions. Include less common shapes such as parallelograms, trapezoids, and ellipses. Also allow students to see concave and irregular shapes – not all octagons must be shaped like a stop sign.
· Use correct vocabulary. Avoid calling a sphere a circle or a rectangle a square. However it is beneficial for students to see a square called a rectangle.
· Make sure students see shapes in multiple orientations. The base of a triangle need not be on the bottom. Draw them with bases rotated through multiple angles. Square need to be rotated also to look like baseball diamonds.
Activities for level two:
· Have students separate geometric figures into “have’s” and “have not’s”. After drawing a line down their paper, they can separate quadrilaterals that have parallel sides from those that do not. They can do the same for triangles that have congruent sides and triangles that don’t. Concave and convex are other dichotomous terms.
· Ask students to classify geometric figures into groups using Venn diagrams. A group title of “four congruent angles” would include both rectangles and squares. “Four congruent sides” includes rhombi and squares. This yields an understanding of the characteristics of a square because those two regions of the Venn diagram overlap.
· Project the shadows of three-dimensional shapes by placing them on an overhead projector. This results in a two-dimensional image. A cylinder appears like a rectangle in one orientation and a circle when turned another way. Students can then observe the properties and guess the name of the object.
· Ask students to measure the interior angle sums of triangles, quadrilaterals, pentagons, and so on. Have them draw the diagonals of quadrilaterals and note that some shapes have congruent diagonals, some have diagonals that intersect at right angles, and so on.
· Have students measure the circumference and diameters of circles. Then divide the circumference by the diameter to approximate pi.
Activities for level three
· Building on the previous activities from level two, ask students to draw conclusions and make conjectures about the properties and patterns they discover.
· After students have measured the interior angles of given triangles to note that they add up to 180°, have them construct their own triangles and test their observations. As they see that the patterns hold, ask them to state a conjecture based on their observations. (Inductive reasoning) Do the same with quadrilaterals, pentagons, etc.
· Demonstrate, and have students demonstrate, properties by making compass and straightedge constructions. The construction of a perpendicular line is based on the fact that the altitude of an isosceles triangle is the perpendicular bisector of the base.
· Have students begin to express their thoughts with oral and written language. Ask them to try to explain why a pattern continues or why a property exists. These informal explanations will lead to more formal deductive proofs at the next level. Constantly ask “why” something happens.
Activities for level four
· Many students don’t understand the value of deductive reasoning and its role in mathematics. Because they can inductively conclude their conjectures hold in all cases, they see deduction as cumbersome, redundant, and superfluous. I introduce the need for this type of reasoning using these two examples:
o If I observe that there are no students in the crosswalk as I come to school on Monday, Tuesday, Wednesday, and Thursday, should I inductively conclude that I should drive through the crosswalk with my eyes shut on Friday?
o If you were accused of a crime, would you want the prosecutor to suggest you are guilty based on the fact that you looked like previous criminals, or would you want the case proven beyond reason using logic and facts?
· I then explain that because our world is governed by math, we need to know that it works in all cases. The fact that I can come up with an infinite number of odd prime numbers does not prove that all primes are odd. Indeed, we need only one example (2) to show this is not true. If NASA were launching me into space, I hope someone had proven that all of his or her math was valid!
· Begin formal deductive proofs by showing students how to prove the very conjectures they made inductively in level three.
Activities for level five
· Because of the high level of sophistication, many of these activities are already contained in the course’s text. A good way to introduce the idea of proof by negation is by the example of prime numbers stated above. I also tell my students to prove that the following statement is false: “Everyone in this room is a geometry student.” They need only one example (the teacher) to prove by negation that the statement is false.
· Non-Euclidean geometry can be posited by questioning their conjectures and proofs from Euclidean geometry. For example, if a line has two perpendiculars passing through two separate points, are the lines parallel? It would seem so, but if you and I walk due north from the equator, we will meet at the North Pole, yet parallel lines never intersect. Additionally the triangle formed by the equator and our pathways has an interior angle sum greater than 180°. This is because the surface is curved as opposed to Euclidean or “plane” geometry.
· When students realize that very little of our world occurs on perfectly flat surfaces, the need to understand non-Euclidean geometry becomes more apparent.
The van Hiele’s noted that students climb through the five levels of geometric knowledge outlined here.
1. Visualization – Children can identify shapes based on appearance, not on properties. Students at this level may not see a square as a type of rectangle nor even recognize it as a square if it is rotated slightly.
2. Analysis – At this level, students begin to associate properties with their shapes. The student who struggled to identify a rotated square will now see that it has four congruent sides and four right angles and is therefore a square.
3. Abstraction – Now students can begin to think about the properties and apply them to arguments that involve inductive reasoning. The student who sees that four different triangles all have an interior angle sum of 180° would use that pattern to reason that all triangles must have the same interior angle sum.
4. Deduction – At this level, students use deductive logic to prove their conjectures from the previous level.
5. Rigor – This goes beyond the former level to explore proofs by negation and non-Euclidean geometry.
Activities for level one:
· Provide students with physical examples of two- and three-dimensional shapes.
· Expose students to a wide variety of geometric shapes in both two and three dimensions. Include less common shapes such as parallelograms, trapezoids, and ellipses. Also allow students to see concave and irregular shapes – not all octagons must be shaped like a stop sign.
· Use correct vocabulary. Avoid calling a sphere a circle or a rectangle a square. However it is beneficial for students to see a square called a rectangle.
· Make sure students see shapes in multiple orientations. The base of a triangle need not be on the bottom. Draw them with bases rotated through multiple angles. Square need to be rotated also to look like baseball diamonds.
Activities for level two:
· Have students separate geometric figures into “have’s” and “have not’s”. After drawing a line down their paper, they can separate quadrilaterals that have parallel sides from those that do not. They can do the same for triangles that have congruent sides and triangles that don’t. Concave and convex are other dichotomous terms.
· Ask students to classify geometric figures into groups using Venn diagrams. A group title of “four congruent angles” would include both rectangles and squares. “Four congruent sides” includes rhombi and squares. This yields an understanding of the characteristics of a square because those two regions of the Venn diagram overlap.
· Project the shadows of three-dimensional shapes by placing them on an overhead projector. This results in a two-dimensional image. A cylinder appears like a rectangle in one orientation and a circle when turned another way. Students can then observe the properties and guess the name of the object.
· Ask students to measure the interior angle sums of triangles, quadrilaterals, pentagons, and so on. Have them draw the diagonals of quadrilaterals and note that some shapes have congruent diagonals, some have diagonals that intersect at right angles, and so on.
· Have students measure the circumference and diameters of circles. Then divide the circumference by the diameter to approximate pi.
Activities for level three
· Building on the previous activities from level two, ask students to draw conclusions and make conjectures about the properties and patterns they discover.
· After students have measured the interior angles of given triangles to note that they add up to 180°, have them construct their own triangles and test their observations. As they see that the patterns hold, ask them to state a conjecture based on their observations. (Inductive reasoning) Do the same with quadrilaterals, pentagons, etc.
· Demonstrate, and have students demonstrate, properties by making compass and straightedge constructions. The construction of a perpendicular line is based on the fact that the altitude of an isosceles triangle is the perpendicular bisector of the base.
· Have students begin to express their thoughts with oral and written language. Ask them to try to explain why a pattern continues or why a property exists. These informal explanations will lead to more formal deductive proofs at the next level. Constantly ask “why” something happens.
Activities for level four
· Many students don’t understand the value of deductive reasoning and its role in mathematics. Because they can inductively conclude their conjectures hold in all cases, they see deduction as cumbersome, redundant, and superfluous. I introduce the need for this type of reasoning using these two examples:
o If I observe that there are no students in the crosswalk as I come to school on Monday, Tuesday, Wednesday, and Thursday, should I inductively conclude that I should drive through the crosswalk with my eyes shut on Friday?
o If you were accused of a crime, would you want the prosecutor to suggest you are guilty based on the fact that you looked like previous criminals, or would you want the case proven beyond reason using logic and facts?
· I then explain that because our world is governed by math, we need to know that it works in all cases. The fact that I can come up with an infinite number of odd prime numbers does not prove that all primes are odd. Indeed, we need only one example (2) to show this is not true. If NASA were launching me into space, I hope someone had proven that all of his or her math was valid!
· Begin formal deductive proofs by showing students how to prove the very conjectures they made inductively in level three.
Activities for level five
· Because of the high level of sophistication, many of these activities are already contained in the course’s text. A good way to introduce the idea of proof by negation is by the example of prime numbers stated above. I also tell my students to prove that the following statement is false: “Everyone in this room is a geometry student.” They need only one example (the teacher) to prove by negation that the statement is false.
· Non-Euclidean geometry can be posited by questioning their conjectures and proofs from Euclidean geometry. For example, if a line has two perpendiculars passing through two separate points, are the lines parallel? It would seem so, but if you and I walk due north from the equator, we will meet at the North Pole, yet parallel lines never intersect. Additionally the triangle formed by the equator and our pathways has an interior angle sum greater than 180°. This is because the surface is curved as opposed to Euclidean or “plane” geometry.
· When students realize that very little of our world occurs on perfectly flat surfaces, the need to understand non-Euclidean geometry becomes more apparent.
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