Do you notice students struggling with geometric proofs and theorems in higher grades and many of them rote learn these proofs and reproduce them in the exam without having a proper understanding of the same?
In the class 9 and 10 Mathematics curricula, around 25 – 30% of the course structure and the assessment blueprints focus on the topic ‘Geometry’. Students are expected to have a deeper understanding of properties of basic geometric shapes and apply the same to understand formal proofs by using deductive reasoning and problem solving. Do higher grade students possess appropriate geometrical knowledge to learn the curriculum objectives? One may assume that since the students have successfully passed the previous grades, they would have the relevant knowledge to learn these concepts. This article examines the idea related to students’ understanding of geometrical thinking.
Van Hiele model of geometrical thinking
In the 1960’s, Dutch educators Dina Van Hiele-Geldof and Pierre Van Hiele were concerned about the difficulties their students were having with geometry and conducted research aimed at understanding children’s levels of geometric thinking to determine the kinds of instruction that can best help children.
According to this theory, there are five levels of thinking or understanding in geometry:
Level 0: Visualization (Basic visualization or Recognition)
Students at the visualization level think about shapes in terms of what they resemble. They are able to sort shapes into groups that look alike to them in some way.
Level 1: Analysis
At this stage, students can list all of the properties of a figure but don’t see any relationships between the properties, and don’t realize that some properties imply others.
Level 2: Abstraction (Informal deduction or Ordering or Relational)
Students at the informal deduction level not only think about properties but also are able to notice relationships within and between figures. They create meaningful definitions. They are able to give simple arguments to justify their reasoning.
Level 3: Deduction (Formal deduction)
Students at the formal deductive level think about relationships between properties of shapes and also understand relationships between axioms, definitions, theorems, corollaries, and postulates. They understand how to do a formal proof and understand why it is needed.
Level 4: Rigor
Students at the rigor level can think in terms of abstract mathematical systems.
(Figure: The Van Hiele theory of geometric thought; Source: Elementary and middle school mathematics (Van de Walle, 2016))
The Van Hiele levels are sequential and to move from one level to the next, children need to have many experiences in which they are actively involved in exploring and communicating about their observations of shapes, properties, and relationships. Many countries (former Soviet Union countries, United States of America, South Africa etc.) have revised their geometry curriculum based on Van Hiele theory.
The Grade 9 and 10 geometry curriculum expects students to start at Level 2 in terms of the geometrical thinking and reach Level 3 by the end of it. It is important for teachers to understand the students’ level of geometrical thinking and adjust the instruction accordingly. The following section gives some insights on the level of students’ geometrical thinking in different grades.
Students Van Hiele levels of geometrical understanding:
The question below tests the students’ ability to identify a straight line.
Only 21% of grade 5 students are able to identify a straight line in different orientations and all the remaining 79% students seem to have some visual perception that a straight line is a horizontal or a vertical line. All these students are at Van Hiele Level 0 in terms of their geometrical thought.
In Q2 shown below, around 38% of grade 6 students are at Level 0. They chose any shape that visually looks like a triangle as a triangle.
If almost 40% of the students are at Level 0 in grade 6, how does it affect the understanding of the grade 7 concepts?
Let’s check the following two assessment items from grade 7. Both the questions are based on properties of straight angles. In Q3 students have to identify that the angles measuring 35o, x o and 65o form a straight angle and find x as (180 – (65+35)) or 80 whereas in Q4 they need to see that the angles measuring x o and 45o make a straight angle and hence x = 180 – 45 or 135 o.
Though both the questions are based on the same property, there is a 15% difference in the correct answer performance. This is because of the difference in students’ level of geometrical thinking. The students who are at Level 0 could identify straight angle only if the straight line is given horizontally and answer Q3 but not Q4. In Q4, some of these students who are at Level 0 or Level 1 have applied the concept of virtually opposite angles incorrectly by thinking that x and the angle opposite to it are virtually opposite angles, again based on a mental image that the vertically opposite angles look like this rather than with the actual understanding that the virtually opposite angles are formed by the intersection of 2 straight lines.
In Q5 below, 45% of grade 7 students have selected an isosceles triangle based on their visual perception which means these students are at Level 0 and Level 1. Students who chose Triangle 3 mostly have this visual image that an isosceles triangle looks like this figure with two equal sides adjacent to the base.
Q6 tests the students’ ability to see the relation between two different classifications of triangles. They should have an understanding that the same triangle can be an obtuse angled or a right triangle AND an isosceles triangle as well. Only 56.6% of grade 7 students answered this question correctly. These students have reached Level 2 stage which is abstraction stage.
Around 14% of the students think that the statements 1,2 and 3 in the above question (Q6) are correct but statement 4 is incorrect. This means that they are unable to see that the same triangle can be classified into 2 different categories though they know the properties or the definitions of different types of triangles. These students are at Level 1. the remaining 30% of the students who are unable to recognize each triangle even with one type of triangle are at Level 0 or at Level 1 (knowing some of the properties but not all).
Q7 below tests the understanding of a trapezium. Students are expected to apply the properties of a trapezium to answer this question.
47% of the students who chose option C have a visual image of a trapezium and they have answered based on that instead of applying the definition of the trapezium. If they apply the definition that a trapezium is a quadrilateral with one pair of parallel sides, they would have selected Shape 4 also.
Based on the above data, it seems that more than 50% of the students are still at Level 0 or Level 1 even in grades 7, 8 or 9 when the curriculum expects them to be at Level 2 by the start of grade 9. By carefully assessing the students’ level of geometrical thinking, the teachers should customize the instructions or the activities to help them move from one level to the next level. A few such activities are discussed below.
Instructions at Level 0: Students should get more opportunities to see shapes in different orientations. For example, students of grades 3 and 4 can have activities like sorting shapes and classifying them into different shapes like triangles, rectangles, squares and circles etc. where the shapes contain different examples in familiar and non-familiar orientations. Similarly, at grade 6 level, while classifying the triangles based on sides and angles, students can be asked to create triangles with different measures using digital tools like Geo-Gebra, Desmos, geometry sketchpad etc. so that they are not limited to the familiar orientations given in the textbooks. To move them to Level 1, they can be asked to create an obtuse-angled triangle with 2 equal sides or to create an isosceles triangle with one of its angles as right angle.
A sample activity from Mindspark on identifying triangles in different orientations:
Grade 3 or 4:
Instructions at Level 1: At this level students should be able to identify shapes based on their properties. For example, the teacher could ask them to sort all quadrilaterals with all 4 sides equal where they should be able to classify both square and rhombus under this property. From the sorted shapes, students can be asked to sort shapes with all angles as right angles. To move them from Level 1 to Level 2, encourage them to see the inclusive relations between different classifications. For example, encourage them to reason out that a rectangle has both pairs of opposite sides parallel and hence it is a parallelogram and all properties of parallelogram hold true for a rectangle OR ask them to justify if all squares are rhombus etc.
A sample item from Mindspark:
Instructions at Level 2: Students should be able to make some informal deductions and understand proofs. Ask students to sort shapes based on properties like ‘all quadrilaterals with diagonals perpendicular to each other’ and let them reason out that both square and rhombus comes into that category and hence it is not the sufficient condition to define either a square or a rhombus. At this level, as students know the individual properties of each shape, they should be able to come up with the necessary and sufficient condition to define each quadrilateral. At grade 7, students can be given the questions like “can an isosceles triangle have 2 obtuse angles?” or “are all right angled triangles scalene?” etc. and encourage them to come up with an informal proof to justify their answer.
Sample items from Mindspark:
To summarise, the Van Hiele levels of geometrical thought, the actual levels of students based on assessment data and the instructions that can be given to students at different levels to help them move from one level to a different level are discussed in this article. Teachers can explore their students’ level of understanding by carefully posing appropriate questions and customize their instructions or activities accordingly so that more students will be at Level 2 by grade 9 and they will be able to deduce the geometrical proofs with correct reasoning.
Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2016). Elementary and middle school mathematics. Pearson Education UK.