Essential Questions What does conceptual understanding mean? What activities can we use to promote...

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Essential Questions

• What does conceptual understanding mean?

• What activities can we use to promote geometric thought?

• What research is available to support conceptual understanding?

What is Conceptual Understanding?

What is the difference between conceptual understanding and

procedural fluency? What teaching methods would promote

conceptual understanding?

What is Conceptual Understanding?

• John Van de Walle suggests that conceptual understanding is when children have an intuitive understanding of mathematics, which can be developed through a task and connect new knowledge to existing knowledge through opportunities to examine different strategies for problem solving.

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Continuum of Understanding

ProceduralUnderstanding

ConceptualUnderstanding

Continuum of Understanding

Find the Perimeters of the Same Area

• What happens to the perimeter when you arrange the same set of blocks to make different shapes?– At your table find all the different shapes you can

make using 1 yellow hexagon, 2 red trapezoids, 3 blue parallelograms & 4 green triangles

– Each shape must share 1 FULL UNIT of length– Find the perimeter of each shape using the edge of

the green triangle as the unit of measure– Record your findings

CONCEPTUAL UNDERSTANDING…

Students with conceptual understanding know more than isolated facts and methods.

Article Conversations

• Read pages 188 -189

In your own words, what are the guidelines for helping create conceptual understanding?

How could these guidelines impact your classroom?

Article Conversations

Read pages 189 - 192o ½ of your table focus on: Similarities in

Ms. Carter’s and Ms. Andrew’s classrooms.

o 0.5 of your table focus on: Differences in Ms. Carter’s and Ms. Andrew’s classrooms.

Whole Group Discussion

Give examples where the teacher promoted conceptual

understanding and why?

What is NEW in the program of studies in

Shape and Space?

Mathematical Processes

[C] Communication

[CN]Connections

[ME]Mental Mathematics and Estimation

[PS]Problem Solving

[R]Reasoning

[T]Technology

[V]Visualization

Philosophy is Constructivism


Intrinsically rewarding since it builds on a natural desire to make sense of things.

3-D Celebrity Heads

3-D Celebrity Heads

Students will identify a 2-D shape or picture of a 3-D shape by asking questions about it.

Curriculum Outcomes3-D Celebrity Heads

C, CN, PS, R, V

5.7 Identify and sort quadrilaterals, including rectangles, squares, trapezoids, parallelograms, rhombuses according to their attributes. [C, R, V]

6.5 Describe and compare the sides and angles of regular and irregular polygons. [C, PS, R, V]

Celebrity Heads

• At each table there is one headband with a shape on it.

• Need a volunteer – NOT TO PEAK at the shape and put it on their heads and stand up.

• The volunteer will then ask yes/no questions of the group to try to identify their shape. They are not to use the name of the shape in their question but they must ask about its sides (eg. parallel, intersecting, etc) and angles (right, obtuse, acute).

Next Steps

• What “next step” in the lesson could help promote connections and build conceptual understanding?

(Celebrity Heads Blackline Master)


Helps students develop positive attitudes about their ability to learn and understand mathematics.

Where are you?

CATCA Teacher’s Convention

Quickly draw a triangle

Quickly draw a hexagon

• Did your triangle have a horizontal line?

• Was the vertex above the line?

• Was there a vertical line in your triangle?

• Was there a right angle in your triangle?

• Was your triangle close to being equilateral?

• Was your triangle almost isosceles?

Did you draw the standard, prototypical triangle?

- standing on its base- equilateral or isosceles

Did you think about other possibilities?

What are the properties of a triangle?

Are any of these triangles?

Why? Why not?

1

2

3

4

56

• Did your hexagon have a horizontal line?

• Was there a vertical line in your hexagon?

• Was your hexagon shaped like the beehive hexagon?

• Did your hexagon have a right angle in it?

• Did it have two right angles? Three?

• More than three?

Our Hexagon

Students’ perceptions of a shape are often based on a limited number of examples.

They are often unfamiliar with nonstandard examples (e.g., textbooks often draw shapes with horizontal baselines).

Students should have exposure to a large number of examples of the various shapes in the form of pictures, manipulatives and real-world situations.

Nonexamples should also be considered.

Van Hiele’s Geometric Thought

Van Hiele’s Geometric Thought

Level 0: Visualization: The products of thoughts at 0 are groupings of shapes that look “alike”Level 1: Analysis: The products of thought at level 1 are the properties of shapesLevel 2: Information Deduction: The products of thought at level 2 are relationships among the

properties of geometric objects.Level 3 Deduction: The products of thoughts at level 3 are looking at relationships between properties of geometric objects and want to know why they operate that way.Level 4: Rigor: The products of thoughts at level 4 are

comparisons and contrasts among different axioms in systems of geometry

Van Hiele Levels

o Level 0: Visualization o identify and operate on shapes according to

appearance. o use the idea of congruency of visual properties

and identification is based on these visual properties, such as “it is a cube because it looks like a box” or “it is a rectangle because it looks like a door.” At this stage, little attention is given to properties of the shapes. In research done in the United States, it was found that at least half the Grade 6 children were still operating at Level 1.

Van Hiele Levels

Level 1. Analysis

• Begin to identify properties of shapes in individual classes and learn to use appropriate vocabulary

• Do not make connections between different shapes and their properties

Van Hiele LevelsLevel 2. Informal Deduction: • The focus is on relationships within classes rather

than relationships between. A cube is now a cube because it is three dimensional with all faces the same sized squares.

• Students should be operating at this level when they enter high school

• Only 44 percent of students in the US were operating consistently at Level 2 at Grade 9.Other studies have shown that 40 percent of students complete high school below Level 2.

Van Hiele LevelsLevel 3. Deduction.• Logical argument is part of this level.

• Internationally most geometry curriculum strive to attain this level. It varies between countries and states as to whether students are encouraged to attain this level in school.

Characteristics of the Van Heile Levels

1. Levels are sequential

2. Levels are not age dependent in the sense of the developmental stages of Piaget

3. Geometric experience is the greatest single factor influencing advancement through the levels.

4. When instruction or language is at a level higher than that of the student, there will be a lack of communication.

How do we use our knowledge of Van Heile’s levels of

geometric thought to guide teaching for



Provides the basis for acquiring new knowledge and solving unfamiliar problems.

Black box geometry

Black box geometry

The purpose of this task is for the students to identify a 2-D shape or 3-D object by asking questions about it. This will require them to use language to ask about its properties without being able to see the shape or object. This activity strongly incorporates the critical math components of communication, reasoning, & visualization.

Curriculum OutcomesBlack Box Geometry

4.4 Describe and construct right rectangular and right triangular prisms. [C, CN, R, V]

5.6 Describe and provide examples of edges and faces of 3-D objects and sides of 2-D shapes that are parallel, intersecting, perpendicular, vertical and horizontal. [C, CN, R, T, V] [ICT: C6-2.2, P5-2.3]

5.7 Identify and sort quadrilaterals, including rectangles, squares, trapezoids, parallelograms and rhombuses, according to their attributes. [C, R, V]

6.5 Describe and compare the sides and angles of regular and irregular polygons. [C, PS, R, V]

ACTIVITY ONE• Students work in pairs or small groups. One

student reaches into the box, one hand on each side, and picks up a shape. The student describes the shape to the rest of the group

• The other students name the shape and draw it

• The shape is them removed from the box and students compare it to their drawing. The box then passes to the next student.

Black box geometry

ACTIVITY TWO• Students work in pairs or small groups. One

student reaches into the box, one hand on each side, and picks up a shape.

• The group questions the student with their hand in the box trying to draw the object. This forces the students to think about properties and move toward van Hiele L 2.

• Students try to be the first to guess what shape is inside the Black Box

Black box geometry

• Yes.• No.• I don’t understand. Please ask again in

another way.• I don’t know. Please tell me how I can

find that out.

How do you play

Some Types of Properties

2-D• Sides• Angles• Diagonals• Symmetry • Concavity

3-D• Edges• Vertices• Faces• Symmetry

Assessment Suggestions

• As the students are describing the shape, note:– Do they describe the attributes using

informal or formal language?– Do they tell what it looks like rather than

using mathematical terms? – Do they use relevant attributes to describe

a shape or solid?


• When the students are asking questions, notice:

– The kinds of language the student uses and the specificity of the question

– Whether the questions focus on distinguishing attributes or on general attributes

Next Steps

What “next step” in the lesson could help promote connections and build



Supports the retention of knowledge. Procedures and skills learned with understanding are easily recalled and applied.

CUT a shape

CUT a shape

• The purpose of this task is for the students to visualize the symmetry of 2-D shapes and to draw one side of that shape against a fold of paper so that when the paper is cut, the shape is made. The students will create symmetrical 2-D shapes by cutting along the fold of paper so that the fold becomes the line of symmetry.

Curriculum OutcomesCut a Shape

4.6 Demonstrate an understanding of line symmetry by identifying symmetrical 2-D shapes, creating symmetrical 2-D shapes and drawing one or more lines of symmetry in a 2-D shape. [C, CN, V]

5.7 Identify and sort quadrilaterals, including rectangles, squares, trapezoids, parallelograms and rhombuses, according to their properties. [C, R, V]

Cut a Shape

1. Fold your paper in half randomly (no edges matching)

2. Imagine you were going to cut a shape from the folded side so that when you unfold the shape you have it is a square. Cut the shape you imagined.

3. Follow instructions 1 & 2 and create:Rectangle TriangleTrapezoid RhombusPentagon CircleHeart shape Parallelogram


• During the activities there are many opportunities to observe the students’ images of half of the shape and the language they use.

• Note whether they can identify the shapes that they cut

• Ask specifically how they know the shape is that particular one and attend to the properties they name. this is particularly relevant for the quadrilaterals.

Next Steps


Quick Draw

Dr. Grayson Wheatley

Quick Draw

The purpose of the activity is to help students develop the spatial sense and ability to form mental images.

Visualization and Verbalize

Quick Draw

1. Tell the students you are going to show them a shape for only a few seconds. Ask students the make a mental picture of the shape so that you can draw it after I turn off the projector.

2. Turn on projector for 3 seconds3. Draw4. Turn on the projector for 3 more seconds5. Revise their drawing6. Discussion – What did you see? How did

you draw it?

Triangle Sort

• The purpose of this activity is to sort the entire collection into three groups so that no triangle belongs to two groups.

Curriculum OutcomesTriangle Sort

6.4 Construct and compare triangles, including

»Scalene» Isosceles»Equilateral»Right»Obtuse»Acute

in different orientations. [C, PS, R, V]


Results in students having less to learn because their understanding is created from interrelated facts & principles.

Triangle Sort

• Look at the various triangles on BLM 20.• Sort the entire collection into three groups

so that no triangle belongs to two groups. • Write descriptions of the groupings.

Equilateral Isosceles Scalene

Right

Acute

Obtuse

Next Steps


Marion Small – Big Ideas

What conceptual understandings do the

children have?

Implications for the Classroom

• Value more than just the right answer

• Make connection between different ideas and representations of concepts

• Organized around core ideas

• Teachers require both mathematical knowledge & pedagogical skill

Article Action Ideas• See page 193 of the article for more

ideas…

Exit Card and

Reflection:

Exit Card and Reflection:

• What does conceptual understanding mean?

• What activities would you use in your classroom to enhance conceptual understanding?

• Where do you fit on the continuum?

Essential Questions What does conceptual understanding mean? What activities can we use to promote...

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