Grade 2 Math Morning
Heather Zetterberg
Math Specialist, Grades K-9
Jessica Murphy
Grade 2 Teacher
Geometry & Visual-Spatial Reasoning
The correlation between spatial skills and mathematical performance
Geometry In Grade 2
Reason with shapes and their attributes
1. Recognize and draw shapes having specified attributes
2. Partition a rectangle into rows and columns of same-size squares
3. Partition circles and rectangles into two, three, or four equal shares
1. Recognize and draw shapes having specified attributes, such as
● A given number of angles
● A given number of equal faces
● Identify:
Triangles
Quadrilaterals
Pentagons
Hexagons
Cubes
Recognize Shapes
Recognize & Draw Shapes
Draw Shapes
Draw Shapes
2. Partition a rectangle into rows and columns of same-size squares
Divide the rectangle into four columns and four rows of same-sized units.
How many small units did you make? _______ units
3. Partition circles and rectangles into two, three, or four equal shares
● Describe the shares using the words halves, half of, thirds, a third of, etc.
● Describe the whole as two halves, three thirds, four fourths
● Recognize that equal shares of identical wholes need not have the same shape
The van Hiele Levels of Geometric Thought
● Intentional instruction is needed to move children through several levels of geometric understanding and spatial reasoning skills.
● Children don't think on a formal deductive level
○ Therefore, they can only memorize geometric facts and 'rules'
○ However, memorization does not lead to understanding relationships
Phases to Develop Geometric Thinking
1: Inquiry - Free exploration of materials
2: Direct Orientation - Targeted investigation
3: Explication - Tasks to develop vocabulary
4: Free Orientation - Open-ended tasks
5: Integration - Summarize and apply
Level 2
Phase 5: Integration
Phase 4: Free Orientation
Phase 3: Explication
Phase 2: Direct Orientation
Phase 1: Inquiry
Phase 5: Integration
Phase 4: Free Orientation
Phase 3: Explication
Phase 2: Direct Orientation
Phase 1: Inquiry
Level 1 - Visual
Level 2 - Descriptive
Phase 5: Integration
Phase 4: Free Orientation
Phase 3: Explication
Phase 2: Direct Orientation
Phase 1: Inquiry
Level 3 - Informal Deduction
Tangrams + van Hiele
Phase 2: Direct Orientation
Phase 3: Explication
Tangrams + van Hiele
Phase 4: Free Orientation
What is Spatial Reasoning?
● The ability to
○ Think about objects in both two and three dimensions
○ Visualize the movement of those objects
● Slide, Flip, Turn
○ Deduce patterns
Mental Rotation
Mental Composition
Mental Composition
Mental Paper Folding
Mental Paper Folding
Space and Number are Linked
The mapping of numbers to space is key to how we “do” mathematics
- Number lines
- Cartesian Coordinate System
- Triangular Numbers
- Pythagorean Theorem
- Measurement systems (clock, ruler)
- Time lines
- Graphs, Diagrams, Tables, …
The Importance of Spatial Reasoning
1. Visual Spatial Skills and STEAM
● Architecture, Engineering, Surgery
2. Spatial Ability in Different Areas
● Art, Gymnastics, Dance, Graphic Design, Sports
3. Spatial Intelligence and Math
● A mental geometric object can be measured, moved, and transformed to facilitate geometric calculation and pattern recognition
● A mathematician uses visual spatial reasoning to enhance quantity comparison, arithmetic, and number sense
A Positive Association
Studies consistently find - a positive relationship between spatial reasoning skills and mathematical performance - better spatial skills are a predictor of future math achievement
Underlying Mechanisms
Spatial reasoning allows individuals to - mentally manipulate and visualize objects in space, which is crucial for - understanding geometric concepts - interpreting graphs - solving complex math problems
Brain Activity Overlap
Neuroimaging studies suggest that
-
similar brain regions are involved in processing spatial information and mathematical reasoning
Characteristics Of Visual Spatial Reasoning
1. Visual Spatial Intelligence is Malleable
2. Gender and the Spatial Skills Myth
3. Spatial Ability and Knowledge are Cumulative and Durable
Characteristics Of Visual Spatial Reasoning
1. Visual Spatial Intelligence is Malleable
● People have different preferred cognitive processes and cognitive thinking styles
● Some are verbal thinkers who have a natural inclination to think in words
● Others are visual thinkers:
O Spatial visualizers
O Object visualizers
Characteristics Of Visual Spatial Reasoning
2. Gender Difference in Spatial Skills Myth
● In America, males tended to outperform females on spatial tasks requiring mental rotation using spatial working memory
● One’s attitude and belief in themselves, and in the amount of effort they put forth, can make a huge difference in visual-spatial tasks performance
Characteristics Of Visual Spatial Reasoning
3. Spatial Ability and Knowledge are Cumulative and Durable
Start early
Play often
Ways To Improve Spatial Reasoning Skills
1. Use Spatial Terms Frequently Corner Side
Examples of Spatial Terms
Type of Terms
Shape
Dimensional Adjectives
Spatial Features
Spatial Relations
What
Examples
Mathematical names that describe twoand three-dimensional objects and spaces. Square, circle, sphere, triangle, pentagon
Terms that describe the size of spaces, objects and people.
Terms that describe the features and properties of two- and three-dimensional spaces, objects and people.
Terms that describe the relative positions of space, objects and people.
Large, small, long, short, big, tiny, tall
Straight, bent, curvy, corner, side, line, corner, pointy, sharp, edge
Inside, outside, under, around, corner, on top of, at the bottom of, in front of, behind, diagonal, across
2. Encourage Children To Use Gestures
● Children often learn better when gestures are used by teachers than when speech is used alone.
● When children use gesturing to indicate movements of objects, their visual spatial intelligence also improves.
● This improvement is also detected in children who do not spontaneously gesture but do so after being prompted to.
3. Teach Children How to Visualize
● Use visual imagery to mentally represent an object not physically present.
● This is a powerful skill in spatial learning and problem-solving.
● Ask students to imagine where an object goes or where an object was placed.
4. Play With Blocks and Build Objects
5. Play Non-Jigsaw Puzzles
6. Expose Children to Map Reading
● Map reading can help children acquire abstract concepts of space and the ability to think systematically about spatial relationships that are not otherwise experienced directly in the physical world.
7. Play Spatial Reasoning Games
8. Explore Photography With Your Child
● Visual spatial perspective taking is the ability to imagine how things look from another viewpoint different from one’s own.
● Taking photos of objects at different angles can enhance children’s ability to take on different visual perspectives and recognize changes in scale.
9. Do Origami
● Mental paper folding has long been used to increase mental rotation ability.
10. Learn to Play Music
● Learning to make music can raise spatial-temporal ability.
● Spatial-temporal reasoning is the ability to think of spatial relations that change through time.
11. Make Three-Dimensional Crafts
A Few Studies
1. Xie et al. (2020) recently performed a meta-analysis of 73 studies and revealed a significant and positive correlation between visual–spatial skills and arithmetic ability [r = 0.25, 95% CI (0.21, 0.29), p < 0.001].
2. Agrillo et al. (2003) showed that non-symbolic numerical abilities positively predicted arithmetic achievements measured by addition, subtraction, multiplication, and division.
3. Recent longitudinal studies provided further evidence that visual–spatial ability could predict arithmetic development (Zhang et al., 2014) and arithmetic ability (Lefevre et al., 2010).
4. Gunderson et al. (2012) observed a predictive relationship between young children's mental transformation skill and their number line estimation. Individual differences in mental rotation performance could have arisen as a difference in any number of components: the ability to carry out rotations, to focus on relevant spatial information, or to carry out non-rotational stimulus matching. Similarly, the number line estimation task, where participants are asked to determine the position of a number along a labeled line, could be decomposed into several components as well (e.g., accessing a representation of a number's magnitude when cued by its symbol, ordering those magnitudes precisely on a continuous number line, spatially subdividing the line at salient landmarks, Siegler and Opfer, 2003)
Resources
● The Development of Spatial and Geometric Thinking: the Importance of Instruction
● The van Hiele Model of Geometric Thinking
● van Hiele Model of Geometric Thinking The van Hiele Levels Phases of Learning
● 12 Easy Activities To Boost Kids' Visual Spatial Intelligence
● Elementary teachers’ attitudes and beliefs about spatial thinking and mathematics
● Seeing as Understanding: The Importance of Visual Mathematics for our Brain and Learning
● What explains the relationship between spatial and mathematical skills? A review of evidence from brain and behavior
● Examining the role of spatial skills and mathematics motivation on middle school mathematics achievement
● The impact of an intervention program on students’ spatial reasoning: student engagement through mathematics- enhanced learning activities
● The Connection Between Spatial and Mathematical Ability Across Development