At a Glance...

In grades 1 to 8, the elements of measurement and geometry are integrated into the Spatial Sense strand. This allows students to develop and extend the skills and concepts inherent in measurement and geometry. Mathematics instruction in these grades should foster strong connections between essential learning related to measurement and geometry (for example, determining measurable attributes of quadrilaterals or prisms).

Source: translated from Guide d'enseignement efficace des mathématiques de la 7^e à la 10^e année, Mesure et géométrie, Fascicule 3, p. 6-7.

Levels of Geometric Thinking

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Image Categories of geometric shapes. Level zero: visualization. Level one: analysis. Level 2: informal deduction. Level 3: deduction. Level 4: rigor. Each level is accompanied by a note card. Starting from level zero, geometric shapes, categories of geometric shapes, one, properties of geometric shapes, 2, link between geometric shapes, 3, deductive system of properties, 4, analysis of the deductive system.

Source: translated from Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, Géométrie et sens de l'espace, Fascicule 1, p. 12.

Two Dutch researchers, Dina Van Hiele-Geldof and Pierre Van Hiele, have devised a five-level model to describe the understanding of geometric concepts at different stages of a student's thinking development. Students can progress from one level of thought to another insofar as they do activities aimed at comparing and classifying geometric shapes and objects, as well as analyzing their properties. Teachers who recognize students' levels of thinking, based on certain observable behaviors, are better able to help them perform geometric analyses and gradually develop their ability to reason.

A brief description of these five levels, as well as observable behaviours and examples for each, are presented in the following table.

The improvement in students' ability to argue comes from the experience they gain in discovering and exploring geometric properties of shapes and objects, organizing them, comparing them with others, and associating them with specific vocabulary. As a result, a student might reach different levels of thinking depending on the learning that has taken place for a particular object, so teachers should not determine a level of thinking in geometry that corresponds to a student's body of knowledge, but rather to more specific learnings.

Each level of thinking in geometry is associated with a specific language. The same word may evoke particular ideas in students in different grades. The word square, for example, conveys certain information to an elementary student who engages in an activity that asks him or her to classify geometric shapes of the same family. However, the intermediate student can recognize that the diagonals in a square are the same length and intersect in the middle, allowing him or her to solve a problem based on these characteristics. (Adapted from Braconne-Michoux, 2014)

In the intermediate grades, the goal is to move students forward, but achieving level 2 is desirable in every grade. This goal is the same as developing the reasoning process, whereby students can deduce, justify, and conclude.

Source: translated from Guide d'enseignement efficace des mathématiques de la 7^e à la 10^e année, Mesure et géométrie, Fascicule 3, p. 20-23.

Measurement

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In measurement, developing a sense of measurement is the goal, just as developing a sense of number is the goal in numeration. A sense of measurement includes the following three components: conceptual understanding, relationship understanding, and procedural understanding. There is a close relationship between these three components.

Image Venn diagram made up of three circles. The shared section of the three circles is the sens of the measure. Circle one: comprehension of the concepts. Circle 2: comprehension of the relations. Circle 3: comprehension of the procedures.

• Students are expected to have a conceptual understanding of the measurable attributes of an object as well as the fundamental concepts that give meaning to units of measurement and the act of measuring. In intermediate grades (grades 7-8 ), the measurable attributes studied are length, area, volume, and angles of two-dimensional shapes and three-dimensional objects. The fundamental concepts related to these attributes are iteration, structure associated with units of measurement, transitivity, conservation, and additivity.
• Students will be able to establish a variety of relationships between standard units of measurement and between attributes. They will be able to determine and compare measurements of two-dimensional shapes and three-dimensional objects. Establishing these relationships will facilitate conjectures and generalizations.
• Students will develop fluency in a variety of procedures related to the multi-step act of measuring. The act of measuring is a collection of thoughts, decisions, and actions that result in an accurate measurement.

Learners entering the Intermediate Division have already acquired some background in measurement. They have explored various measurable attributes of objects and the fundamental concepts that underlie them. They have also established some relationships between standard measurements and have developed various skills in measuring accurately.

Measurement and Geometry

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Measurement and geometry are at the heart of our daily activities and, in so doing, help us understand the world around us. Observing and describing an object requires mastering concepts related to these two fields. The applications of measurement in geometry are many and varied, whether it is to explain the shape of a soap bubble, determine the popularity of a web page, or estimate the population in certain districts. Measurement involves making explicit the conjectures made when solving problems. For example, determining the importance of a web page is as simple as relying on the number of links associated with other web pages.

The use of measurements in geometry helps determine unknown length, area, or volume using known measurements of two-dimensional shape and three-dimensional object representations. Measurements can be calculated from attributes such as side length, surface area, or the coordinates of a shape in a Cartesian plane.

• To construct and measure the perimeter or area of complex (compound) shapes, students need opportunities to decompose shapes or rearrange their parts into other simple shapes such as circles, polygons, or other.
• To calculate the perimeter, area, or volume of three-dimensional objects, students need to complete activities that allow them to decompose objects and rearrange them by constructing spheres, cones, pyramids, or prisms, and even by otherwise assembling parts of objects. Activities for estimating measurements of three-dimensional objects provide students with a useful strategy for calculating the dimensions of irregular objects.

Example

Image Rectangular paper prisms and stacking cubes are placed side by side. Prism “ A “ and prism “ B “ each in paper and cube form.

In short, certain essential concepts support the learning of measurement integrated into geometry.

• Calculate the area and volume of a geometric shape or object by breaking it down into simpler geometric shapes or objects or rearranging its parts.
• Estimating measurements using a variety of tools, including technology, supports the development of measurement skills.
• Calculate the perimeter, area or volume of the image of a known shape or object following a transformation, using ratios and proportions.

Source: translated from Guide d'enseignement efficace des mathématiques de la 7^e à la 10^e année, Mesure et géométrie, Fascicule 3, p. 54-55.

Ability to Visualize

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The ability to visualize is the ability to form a mental image of a situation or an abstract concept. In measurement, this ability is mainly related to the ability to form a mental image:

• of certain measurable attributes;
• benchmarks associated with the various attributes.

Visualize certain measurable attributes: the ability to form a mental picture of certain measurable attributes helps students better understand their meaning. In the Intermediate Division, students should develop the ability to visualize the attributes volume, length, area, and angles.

Visualize benchmarks: the ability to form a mental image of some of the benchmarks associated with measurable attributes helps students estimate the magnitude of an attribute or check the reasonableness of a result obtained from using a measuring instrument or applying a formula.

Source: translated from Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, Mesure, p. 20-22.

In the intermediate grades, students continue to investigate the properties of two-dimensional shapes and three-dimensional objects through representation on paper (for example, representation on graph paper, dot paper).

These activities help students construct a mental representation of two-dimensional shapes and three-dimensional objects based on specific properties and promote the development of visualization skills.

In order to construct a good mental representation of three-dimensional objects, students must be able to visualize them both in the two-dimensional space of two-dimensional shapes and in the three-dimensional space of three-dimensional objects.

To help them develop the ability to move from one space to another and to consolidate their knowledge of the properties of three-dimensional objects, teachers can use a variety of construction activities and representational activities (for example, tracing the net of a three-dimensional object, associating a three-dimensional object with its net or with its front, side, and top views).

Source: translated from Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, Géométrie et sens de l'espace, Fascicule 1, p. 28-30.

Spatial visualization is developed by mobilizing certain spatial skills listed by John Del Grande (1990), including mental rotation.

(John Del Grande, 1990. © 2019, National Council of Teachers of Mathematics. The Arithmetic Teacher, vol. 37, no. 6.)

Source: translated from Guide d'enseignement efficace des mathématiques de la 7^e à la 10^e année, Mesure et géométrie, Fascicule 3, p. 12-13.

Ability to Solve a Problem Situation

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The ability to solve problems is an essential process in learning measurement. To help students develop this skill, teachers must present them with various types of problem situations in a meaningful context. It must encourage them to call upon their previous knowledge as well as their literacy and problem-solving strategies, to clearly communicate their results and to discuss the ideas of their peers during mathematical conversations. By being thus engaged in a reflection on the targeted concepts, the students will clarify the concept’s meaning.

Measurement problem situations should contribute to students' understanding of fundamental attributes and concepts and relationships. It is essential that students be actively involved in problem solving and subsequent discussions. These varied experiences allow them to develop their sense of measurement.

Source: translated from Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, Mesure, p. 23.

Reasoning Skills

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The following is an example of specific expectation that supports the development of spatial reasoning, taken from the Ontario Mathematics Curriculum, Spatial Sense, Grade 7 :

- Use the relationships between the radius, diameter, and circumference of a circle to explain the formula for finding the circumference and to solve related problems.

In a mathematics class focused on the development of reasoning, teachers will also focus on geometric reasoning. This expectation targets, in particular, the development of skills related to deduction and the presentation of coherent mathematical arguments. Teachers who ask students why their statement is true should observe how they approach the question to help them progress in acquiring geometry skills. According to research on geometric thinking (Van Hiele, Uziskin, Burger and Shaughnessy, and Braconne-Michoux), the way students justify their geometric reasoning shows their level of geometric thinking. The student who, for example, justifies that a triangle is isosceles, because it looks like a pointed hat, will eventually have to verbalize this characteristic by saying that at least two of the sides of the triangle or two of its angles are congruent.

Source: translated from Guide d'enseignement efficace des mathématiques de la 7^e à la 10^e année, Mesure et géométrie, Fascicule 3, p. 20.

Communication Skills

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Oral or written communication is a key element in ensuring students' deep understanding of concepts. The teacher's role is to facilitate mathematical communication during exploration and during small-group or whole-class mathematical exchanges. The teacher encourages students to share their hypotheses with others and to justify them with convincing arguments and appropriate vocabulary. In addition, the teacher should encourage students to reflect on their thinking during exchanges and consolidation. Discussions should be guided in such a way as to allow students to make connections between different representations and mathematical concepts.

Source: translated from Guide d'enseignement efficace des mathématiques de la 7^e à la 10^e année, Mesure et géométrie, Fascicule 3, p. 36.

Ability to Abstract

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In the primary grades, students' geometric thinking is generally at levels 0 and 1. At first, students learn to recognize geometric shapes and objects, name them and list some of their physical attributes (for example, the sphere is round and rolls). Many then construct a mental representation of geometric shapes and objects by associating them with known objects (for example, a rectangle looks like a door, a sphere looks like a balloon). In some cases, this representation is so entrenched that it prevents them from conceiving of an orientation of the shape or object other than the one that corresponds to their representation. See the figures below.

Image Students associate a rectangle to a known object, for example, a door. The text is accompanied by a drawing of a rectangle and its square angle. This figure doesn’t correspond with the image of a rectangle that certain students imagine. The text is accompanied by a drawing of a rectangle and its square angle in an oblique posture.

Later, students gradually move away from references to physical attributes and known objects as they learn to describe geometric shapes and objects in terms of a list of necessary properties (for example, a rectangle is a two-dimensional shape that has four sides, four right angles, and opposite sides that are congruent and parallel).

Image A rectangle with dashes on the parallel sides and the square angles. A dash, first pair of congruous sides. 2 dashes, second pair of congruous sides.

In the junior and intermediate grades, students' geometric thinking progresses to level 2 as their capacity for abstraction develops. This ability is related to the ability to focus one's attention on certain properties in isolation. It allows them, among other things, to choose from the list of necessary properties of a geometric shape or object, those which are sufficient to define it (sufficient properties).

Source: translated from Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, Géométrie et sens de l'espace, Fascicule 1, p. 32-33.

It is important for teachers to be aware that in order to develop skills and concepts related to measurement and geometry, students need to develop some spatial skills. However, according to Focusing on Spatial Reasoning K-12 (Ontario Ministry of Education, 2014), 'not all types of skills inherent in spatial thinking are related to mathematics outcomes to the same extent. Learning mathematics depends more on some spatial thinking skills than others.' Therefore, teachers need to help students particularly develop their spatial visualization and visual-spatial memory.

It is clear that constructing or describing geometric shapes and objects requires spatial reasoning; however, teachers must also realize that students use strategies to estimate that require spatial thinking and, in particular, spatial visualization.

While taking into account the elements specific to the development of spatial reasoning, teachers must integrate the essential elements of effective teaching of mathematics, in particular the skills related to communication and problem solving.

Image Infographic. A gear is placed with an axis called “ resolution of problems : The wheel of the gear, which is on his axis is divided in three parts: the reflection; the selection of appropriate tools and calculation strategies; establishing links; modeling; and reasoning. The whole revolves around communication. These five elements allow: the thought abilities, application and communication. On the gears, modelling and reasoning, we find another wheel of conceptual comprehension, and a wheel of procedural comprehension. This will breed knowledge and comprehension.

Teaching According to the Levels of Geometric Thinking

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According to Van de Walle and Lovin (2008), the geometry explorations that teachers provide to students are the primary factors that determine learners' progress along the geometric thinking developmental scale. Students progress through the various levels of geometric thinking to the extent that they engage in exploration, manipulation, construction, visualization, comparison, and transformation activities. Exploration activities should enable them to make connections between geometric concepts in order to construct meaningful ideas. In addition, the way in which they are interacted with can encourage or at least challenge them to move to the next level.

The levels of geometric thinking describe students' ideas about geometry and how they perceive shapes and objects. Each level is defined as a sequence of logical behaviors. As students progress from one level to the next, they change their thinking as they describe, perceive or compare geometric shapes and objects.

There are other elements involved in the development of levels of geometric thinking.

• The levels are sequential; to move to a higher level, students must develop ideas and understanding related to a geometric concept from the previous level.
• The levels of geometric thinking do not depend on students' age or grade. Some adults are at level 0. Students may be at level 1 (analysis) in understanding one concept and at Level 0 (visualization) in mastering another concept. For example, they describe some properties of the square (level 1: analysis), but recognize the parallelogram only because of its appearance (level 0: visualization).
• Students move from one level to another based on their understanding of the various concepts.
• Students who are in level 2 have acquired levels 0 and 1.
• The factors that most influence the development of geometric thinking are activities that are appropriate to the level of geometric thinking of the students. The level of geometric thinking of learners depends on the concepts taught and the strategies used.

Note: The five levels of geometric thinking development described by Van Hiele and Van Hiele-Geldof are in no way related to the four levels of achievement in the mathematics curriculum achievement chart.

Teaching at Level 0 (Visualization)

At level 0, teachers should provide students with many opportunities to classify geometric shapes and objects. Emphasis should be placed on the similarities and differences between geometric shapes and objects.

A key idea in geometry is that of invariance. A significant amount of time spent studying geometry should be spent determining what changes are possible for a property to remain true. For example, the sum of the measures of the interior angles of a triangle does not change, regardless of what changes a triangle may undergo. Ironically, the more constraints teachers place on what can be changed, the more they help students identify interesting properties. For example, when exploring the measurement of interior and exterior angles of polygons, the constraint of limiting exploration to quadrilaterals would allow for a test of whether some of the properties identified apply to convex or concave quadrilaterals.

An additional constraint of limiting the exploration to special cases of quadrilaterals such as parallelograms, kites, and rhombuses would help students determine what is invariant.

Teaching at Level 1 (Analysis)

In level 1, teachers provide students with geometric shape and object construction activities to give them the opportunity to analyze a variety of cases. Dynamic construction activities help students at this level to progress. They can explore the construction of geometric shapes and objects and then group them into different classes based on their characteristics.

Teaching at Level 2 (Informal Deduction)

At level 2, teachers help students develop skills related to informal inference by using appropriate vocabulary. Students develop these skills by gaining experience in understanding statements related to informal deduction; for example, 'If it is ______, then it is also ___.'; 'All ____ are ____.' Statements, such as the ones below, help learners reason and acquire vocabulary specific to informal deduction.

• If it is a rectangle, then it is also a parallelogram.
• All squares are rectangles.
• Some parallelograms are rhombuses.

It is important to encourage students to examine the properties of a class of shapes or objects to determine which properties are necessary and which are sufficient to describe them. A necessary property, if removed from a description, would no longer uniquely define the shape or object in question. For example, removing the property that a square has at least one right angle, and keeping the property that a square has four congruent sides, describes not only the square, but also the rhombus. In order to describe the square, it is therefore necessary to have two properties, the one concerning the length of its sides and the one related to the right angles. In Grade 6, students begin to explore the diagonals of quadrilaterals. They find that describing only the diagonals is sufficient to classify quadrilaterals. For example, students need only say that the diagonals of the shape are the same length and intersect in the middle at a right angle to describe the square. Other properties are no longer necessary.

Students should be encouraged to formulate and test hypotheses, using questions such as 'Why…?' to try to determine why there are particular relationships between certain elements of a geometric shape or object.

Source: translated from Guide d'enseignement efficace des mathématiques de la 7^e à la 10^e année, Mesure et géométrie, Fascicule 3, p. 24-27.