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Find the area and perimeter of this shape?
While waiting to start, consider what is challenging about this problem:
10
Spatial Reasoning &
Mathematics
Thanks to the students and teachers of Math for Young Children, Jackman Institute of Child Study, Diane’s Classes
Diane R.H. [email protected] [email protected]
Zack [email protected] of Toronto
OutlineWhat is spatial reasoning?
Importance of spatial reasoning to education
Improving spatial reasoning in the mathematics classroom: Big ideas
Examples of student thinking illustrating important ideas
Spatial activities across strands and grade levels
Spatial adaptations to common mathematics activities
Summary: Improving spatial reasoning
What is Spatial Reasoning?The ability to create and manipulate mental
representations of actual and imagined shapes, objects, and structures (Cohen & Hegarty, 2012).
Spatial thinking is not a unitary construct and involves many related and unrelated skills, including:Navigation skills Reading maps, graphs, visual data Imagining objects move in space VisualizationManipulating/Creating/Designing/using objects Perspective taking Remembering locations of objects Moving one’s body in space Proportional reasoning Non-verbal reasoning Visual intelligence Packing car Assembling Furniture
Mental Rotation:Are these the same shape?
Mental Rotation:Are these the same shape?
Paper-FoldingAre these the same shape?
Paper-FoldingAre these the same shape?
Why Spatial Reasoning
“Spatial thinking is the principal complement to verbal thinking” (Newcombe & Frick, 2010, p. 102)
Spatial ReasoningIn education, the importance of spatial reasoning has becoming increasingly recognized over the last few years.
For example, spatial reasoning is linked to Understanding graphs and diagrams
Complex hierarchal relations ie taxonomies in biology
Understanding geography and geology
Metaphors in English- close to a goal
Dentistry, policing, working with machinery…
Creating databases
Success and creativity in Science, Technology, Engineering & Mathematics
©DianeTepylo & ZackHawes2014
Spatial Thinking Supports: (Newcombe, 2010; Newcombe & Frick, 2010)Scientific visualizing
Faraday and electromagnetic interactionsStructure of DNAEinstein’s descriptions of his thinking
pracesses
Metaphoric use “Close to a goal”, “an insider”
Everyday tasks : Finding your way Packing car Assembling furniture
Spatial Thinking Supports: (Newcombe, 2010; Newcombe & Frick, 2010)Reading graphs and diagrams
Slope- if students don’t pay attention to orientation, how are they going to see slope
Understanding point of intersection as being the same spot
Maps for thinking Determining the source of cholera in London
in the 1800s
A mathematician sees … and communicates
Plotted the homes of those who died, on a mapAdded the wells to the map
With this map, convinced the parish council to take the handle of the Broad Street Pump.
A cholera epidemic in London, 1854. A doctor suspects cholera is connected to water.
Spatial Reasoning and MathematicsNumerous studies show spatial reasoning predicts mathematics performance (Battista, 1990; Burnett, Lane, & Dratt, 1979; Casey, Nutall, & Pezaris, 1992; Casey, Nuttall, & Pezaris, 2001 ; Casey, Nuttall, Pezaris, & Benbow, 1995; Delgado & Prieto, 2004; Doyle, Voyer, & Cherney, 2012; Geary, Saults, Liu, & Hoard, 2000; Guay & McDaniel, 1977; Kyttälä & Lehto, 2008; Reukala, 2001; Robinson, Abbott, & Berninger, 1996; Tolar, Lederberg, & Fletcher, 2009)
3D spatial tasks were moderately correlated with algebra scores and strongly correlated with SAT-M scores (Tolar, Lederberg, & Fletcher, 2009)
Spatial & Verbal Reasoning in Mathematics
Verbal
Number facts
Naming shapes
Using measurement formulas
Spatial
Magnitude and estimation
Seeing relationships between shapes
A square is a special case of a rectangle
A rectangle is a special case of a parallelogram
Understanding measurement formulas
Geometry & Spatial Reasoning
Spatial reasoning in ninth grade significantly predicted gemetry in 10th grade and 12th grade. Sherman (1979)
Measurement & Spatial Reasoning
(Casey, Dearing, Vasilyeva, Ganley, & Tine, 2011)
Justin’s Story
Measurement & Spatial Reasoning
(Casey, Dearing, Vasilyeva, Ganley, & Tine, 2011)
Spatial sense linked to number sense
Magnitude and spatial reasoning happen in the same areas of the brain
After practice with spatial transformation questions, Grade 1 students scored better on adding and subtraction tests (Mix &
Cheng, 2012)
Especially on questions like 4 + __ = 12
Spatial Reasoning & Number Sense
Spatial Reasoning & Logic/
ClassificationVenn Diagrams
• Consider:• All A are B • All B are C• Therefore All A are C
• Is this language based?
A B C
Slide from W. Whitely from Vinod Goel, Lakoff and Nunez
No
Spatial Reasoning & Logic/
Classification
Spatial Reasoning & Data
management
Spatial Reasoning & Problem
SolvingSpatial reasoning predicts problem-solving ability where problem-solving is described as ill-defined questions that can not be solved simply by a formula (Casey, et al., 1995; Delgado & Prieto, 2004; Hegarty & Kozhevnikov, 1999; Johnson, 1984; Newcombe and Frick, 2010; Sherman, 1979; Delgado and Prieto, 2004)
Effect stronger in higher grades and continues even at PhD level
Spatializing the Mathematics
ClassroomEven with ever increasing evidence of the importance of spatial reasoning, we still need to determine what effective teaching of spatial reasoning looks like in the classroom
Not new expectations, but a new perspective on teaching the content.
From psychological research, we know that There are substantial differences in spatial ability from age 3
training in spatial reasoning leads to durable transferable changes
Some Big Ideas in Spatial Reasoning &
Math1. Visualization
2. Learning to see differently
3. Seeing and representing space
Position, orientation & movement
4. Invariance- recognize what doesn’t change (and what does)
5. Composition & Decomposition- important in measurement
6. Intentionally connecting 2D and 3D
7. Symmetries
8. Perspective taking Working Group: Young Children’s Spatial Reasoning & Mathematics 2012
1. Visualization
Visualization is an important mathematical process
Helps reduce with cognitive load
Helps understand what is going on in problems
Gains more importance when problems are novel or complex
Practicing Visualization : Quick Draw activity
Show picture for 3 seconds
Cover picture- have class draw what was illustrated
Draw the image
1. Quick Draw
1. Discussing visualizations
Think Pair Share
“How did you know what to draw”
What’s My ShapeIt has 5 faces
It has 5 vertices
One face is a square
It looks like a monument found in Egypt.
Shy Graph
1. Visualization
Visualization improves with practice
Most of the activities we will share today have a visualization component
Strategies for visualization tasks
Gestalt
Negative space
Verbalizations
Teaching students to visualize
Visualize, Verbalise, Verify
How much boxboard is
needed to make a case for 12 pop cans in 4 rows?
1. Visualizing & the Number Line
JIS Activity
Place the following values on a number line:
.5, 1/3, 75%, 125%, .01
Add another 3 numbers to the number line.
0 10
2. Learning to see differently
Visualization activities promote a variety of strategies for encoding images and patterns. Flexibility is key as many math problems are easier to solve when one perceives the problem in a specific way:
A Patterning example:How does the number of blocks change from one position to the next?
How many blocks are in the nth position?
A mathematician sees … Learning to see differently
1+2+3+ … n = ?
n
n+1
n 2(1+2+3+ … n)
= n(n+1)
1+2+3+ … n = n(n+1)2
Dan Meyer’s Circle Square Task
2. Learning to see differently:
spaces vs ticks on a ruler
Elapsed Time
2. Learning to see differently
A geometry example
Think- Pair-Share “What all do you see in the drawing below?”
Primary- JuniorSecondary
2. Learning to see differently
Orientation affects perception
2. Learning to see differently:
Working with visuals1. With a partner, discuss everything you can
tell from the diagram. 2. What questions could you ask your students
to help them see more in the graphic?
3. Seeing and Representing Space: Position, Orientation,
Movement
Students need to learn how to
structure space
What Children Pay Attention to?
JK Boy: “They’re the same”
What Children Pay Attention to?
Grade 2 Girl: “They’re the same”
What Children Pay Attention to?
Grade 2 Boy: “They’re the same”
3. Seeing and Representing Space: Position, Orientation,
MovementBeing clear about what matters Until school, kids learn to ignore rotation, orientation
A toy is a toy in any location, from any angle
In school, sometimes orientation mattersFor letters b, p, d
In some geometry tasks orientation matters but not in others
A triangle is a triangle regardless, but
The orientation of a line matters when studying slope
Vertical & horizontal easier for children to followYounger children have difficulty copying/remembering slanted lines unless there is a stable parallel line in background
8 year olds will draw chimneys perpendicular to roof not to ground (Bryant, 2007)
3. Seeing & Representing space:
SecondarySlope- if students don’t pay attention to
orientation, how are they going to see slope
Understanding the difference between two pointsWhile children (8-9) can be taught rudiments of
plotting points on a coordinate systemFinding the difference between 2 points in much
more difficult An Implied transformation
Understanding systems of equations Do students understand that the intersection occurs
is the same point in both equations: x in equation 1 = x in equation 2 y in equation 1 = y in equation 2
3. ‘Look, Make, Fix’
IS JIS
3. Master Builder/Barrier Game: Structuring Space with
GridsIntermediate/Senior (graph paper)
List instructions
Put lines on graph given slope ,
Desmos
3. Transformations
3. TransformationsSlides, flips, reflections
Research suggests many secondary students and adults have difficulty envisioning (Wright et al., 2008)
First mentioned in curriculum in grade 3, continue all through elementary and secondary mathematics
Even though not into curriculum until grade 3, transformations underlies many mathematical activities occuring earlier
1 minute Video (building with blocks, composition and decomposition activity trying to make square?
Transformations – 2/3 Visualize, verbalize, verify
From Seeing to moving to traditional exercisesGrade 2/3 teachers found that transformations were easier with inclusion of visualize and verbalise
Transformations: secondary
Traditional order– graphing calculator
Graph a bunch of equations
Note what changes to equation do
Changing order helped struggling students with a guided discovery
Present two graphs: how are they different (how transformed ?)
Then show equations
“What does changing the equation do to the graph”
Paying attention to
Transformations
Working with visuals
Connect 2D & 3D views
Finding metal required for vent
Summary: Effective Strategies for Improving
Spatial Reasoning From the research:
Mindful practiceSpatial Manipulative Play (blocks, puzzles)Gesture VisualizationWorking with graphics
Drawing, AnimationLanguageProviding spatial frameworks (grids, Cartesian plane)
Summary: Classroom ActivitiesLook, Make, Fix
Master Builder (Barrier games)
QuickDraw
Visualize, verbalize, verify
Pentominoes and polyominoes
Broken ruler tasks
Discuss visualsDon’t assume students see visuals as you do
General thoughts about spatializing
mathematicsBe aware of that perception is largely a top down process
As teachers we easily see certain details because of our deep knowledge
Spend time with the visuals before using themRecognizing transformations implied in diagrams;
Be aware of Incorrect or unconventional interpretation of graphs and visuals;
Don’t assume what you see in the visual is apparent to every one
Thank You More information
Math for Young Children websiteshttp://www.oise.utoronto.ca/robertson/Inquiry-based_Mathematics/Math_For_Young_Children/index.html
Trent Mathematics Education Reseach Center: http://tmerc.ca
Spatial Reasoning and MathematicsContact Diane at [email protected]
