1. Spatial Reasoning Overview
Welcome to today’s overview session on Spatial Reasoning. I am ___________.

2. Learning Focus Participants will:
deepen mathematical content knowledge of spatial reasoning
develop awareness of the key skills associated with spatial reasoning
develop pedagogical knowledge for teaching spatial reasoning
engage in spatial reasoning through mathematical tasks
Our session today is based upon many years of research, including recent research within Ontario by Dr. Joan Moss, Dr. Cathy Bruce, Zach Hawes and Tara Flynn. The research has highlighted some critical learning which we will explore further today. The research has identified that spatial reasoning encompasses many skills, many which are not well understood within education. We will explore the numerous ways that we use spatial reasoning and then we will connect it to specific mathematics tasks. In particular, you will: <<read the bullets>>

3. Session Norms
Be engaged in the tasks and discussions as this enriches everyone’s experience.
Be a learner – it is fun!
Actively seek connections between the research and your classroom experience.
By better understanding student thinking we better understand the impact we can have on their learning.
These are the norms for today’s session.

4. Agenda Activity: Exploring Pentominos Spatial Reasoning Connections
Why Pay Attention to Spatial Reasoning
Turning to Research
Activity: Cube Challenge
Taking Perspective
Connecting to the Curriculum
This is what we will be doing in this session. We will engage in a blend of activities and discussions as we dig into spatial reasoning.

5. Exploring Pentominoes
How many unique shapes can you make?
Instructions:
Use 5 square tiles.
Ensure that every square has at least one side fully touching another square.
<<Note to Presenter: text enclosed by << >> are notes for the presenter and not part of the script>>
Now it is your turn – let’s explore pentominoes. A pentomino is a special shape formed by grouping 5 congruent squares so that every square has at least one of its sides in common with another square.
Working with a partner, consider how many unique shapes you can make if you are using 5 squares and every square must have at least one side fully touching another square.
This task is widely available, including in TIPS4RM Grade 7 Unit 8.
Taking Shape, by Moss, Bruce, Caswell, Flynn and Hawes, further explores the role that this task and others have in building spatial reasoning skills in children.
<<Allow time for participants to work before moving to next slide.>>
<<You may wish to have participants use the Colour Tiles by mathies.ca>>
A pentomino is a shape formed by grouping 5 congruent squares so that each shares at least one of its sides with another.

6. Testing for Uniqueness
Are the shapes on the screen unique? <<no>>
Another way to ask this is “Are these shapes congruent?”
Do they have the same shape and same size?
What test can you use to explain this? <<Is it possible to lay on exactly on top of the other?”
<<Allow time for participants to continue working to find all 12 unique shapes>>
<<If you use the Colour Tiles by mathies.ca you can rotate and reflect the shapes to see if they are unique.>>

7. Did you find all 6?
Take a minute to reflect on or discuss the kinds of spatial reasoning that you may have been doing in that activity. What mathematics were you using?
You may have identified visualization as well as mental rotation and reflection.
As well, for young children, it is easy to see how this kind of activity can help to reinforce “fiveness”.
<<extension if time permits>>
Which of the 12 shapes has the least perimeter?
What do you know about their areas? <<same>>
What about the shapes creates a smaller perimeter? <<interior edges>>
<<If participants use the Colour Tiles by mathies.ca you can change the colours of each pentomino if you would like>>

8. Where is the Math? What strategies did you use?
How did you convince yourself you were correct?
How did knowing there were 12 influence your thinking?
Discuss with a partner your responses to these questions. <<Allow 5 minutes to discuss>>
Who would like to volunteer to share a response? <<Have a whole group discussion to highlight the range of strategies, the spatial language used in the responses, etc.>>

9. What is Spatial Reasoning?
Spatial thinking, or reasoning, involves the location and movement of objects and ourselves, either mentally or physically, in space. It is not a single ability or process but actually refers to a considerable number of concepts, tools and processes. National Research Council, 2006
<<Read the quote>>

10. Spatial reasoning is not a unitary construct and involves many related and unrelated skills, including:
Navigating & wayfinding
Imagining objects move in space (mental rotation and transformation)
Composing / decomposing
Partitioning
Visualizing
Non-verbal reasoning
Scaling up or down
Manipulating objects
Creating or designing objects
Moving one’s body in space
Disembedding
Comparing
Perspective taking
Shifting dimension
Diagramming
Locating objects/ remembering locations of objects
Spatial reasoning is not a unitary construct and involves many related and unrelated skills identified on this slide.
Take a moment and read over the slide. Turn to an elbow partner and identify one skill that you used in the pentaminos task. Identify one skill that you would like to know more about.
<<Allow time for discussion. After everyone has had a chance to talk with their elbow partner, ask volunteers to share out their responses. Record the skills that people want to know more about. This list will be referenced later in the session.>>
Orienting
Creating & reading maps, graphs, visual data
Unitizing (same space & shape)
Proportional Reasoning

11. Spatial Reasoning
These ideas are also captured in the graphic on page 4 of the Paying Attention to Spatial Reasoning document.

12. Why is it important?
“…spatial intelligence has evolutionary and adaptive importance. Any mobile organism must be able to navigate in its world to survive and must represent the spatial environment in order to do so.” Newcombe & Frick, 2010
We are mobile organisms in a 3D world. <<click>> Newcombe and Frick state that “spatial intelligence has evolutionary and adaptive importance. Any mobile organism must be able to navigate in its world to survive and must represent the spatial environment in order to do so.”
<<Image:

13. Why Pay Attention to Spatial Reasoning?
Spatial reasoning is a “major blind spot” in education…”locked in a curious educational twilight zone: extensively relied on across the K-12 curriculum but not explicitly and systematically instructed in any part of the curriculum.” National Research Council, 2006
<<read quote>>
The current work in Ontario, including sessions such as this one, are designed to build understanding of spatial reasoning across the various strands in the curriculum.

14. Revisiting Pentominoes
Now let’s discuss in more depth the spatial reasoning skills you used to complete the task. <<go to next slide>>
What spatial reasoning skills did you use to complete the task?

15. Spatial Reasoning Skills
Navigating & wayfinding
Imagining objects move in space (mental rotation and transformation)
Composing / decomposing
Partitioning
Visualizing
Non-verbal reasoning
Scaling up or down
Manipulating objects
Creating or designing objects
Moving one’s body in space
Disembedding
Comparing
Perspective taking
Shifting dimension
Diagramming
Locating objects/ remembering locations of objects
Orienting
What skills did you use in the pentominoes activity? <<Allow time for whole group discussion. Refer to the list of skills that participants wanted to learn more about. If some of these skills are identified as relating to the pentomino activity, have participants discuss the connections to the task. Some skills that participants may be identified include manipulating objects, imagining objects moving in space (mental rotation and transformation), visualizing, composing/decomposing. In particular, since the mirror images/translations and rotations do not count as additional combinations, these pentomino activities develop an understanding of congruence.>>
Creating & reading maps, graphs, visual data
Unitizing (same space & shape)
Proportional Reasoning

16. Research Summary
Three Reasons to Pay Attention to Spatial Reasoning in Mathematics
Spatial thinking and mathematical thinking (and achievement) are strongly related
Spatial thinking is malleable and can be improved through education and experience
Schools play an important role in fostering spatial thinking
These are three reasons identified in the Paying Attention to the Spatial Reasoning document. <<allow time for people to read them.>> We will now take a few minutes to learn more about this research.

17. Reasons to Pay Attention to Spatial Reasoning
Complete the reading as follows: #1 “Turning to Research…” (bottom of page 5 to the end of 1. Spatial Thinking is critical .... (page 6) #2 “2. Spatial thinking is malleable …” bottom of page 6 to top of page 7 #3 “3. Schools play an important role…” (middle of page 7) and “Gender inequities” (top of page 8) #4 “Mathematics learning difficulties (middle of page 8 to top of page 9)
We are now going to do a jigsaw reading activity. Number off from 1 through 4 at your table. Move to the table that matches your number. This is your expert group.
On your own, complete the reading assignment from the Paying Attention to Spatial Reasoning document listed on the slide.
<<Allow sufficient “quiet time for reading and note taking. (12 minutes)>>
As an expert group, work together to summarize the main points from your reading assignment on chart paper.
When finished, post your summaries and move back to your original table, your home group.
In your home groups, take turns sharing your expert group summaries.

18. Let’s Do Some (MORE) Math!
Spatial Reasoning
Now we are going to do some more tasks which highlight the role that spatial reasoning plays in mathematics learning.

19. 4-cube Challenge Join 4 cubes together.
Make as many different combinations as possible.
One example
Working in pairs, join four cubes together. Make as many different combinations as possible. You should have enough cubes to allow you to save all of your combinations. This will allow you to test for equivalence/congruence.
<<Give participants time to build their combinations.>>

20. Checking for Equivalence
Have you found ALL the different non-equivalent (non- congruent) figures?
How are you deciding which figures are different or non-equivalent? <<Ask participants to describe their reasoning when comparing shapes. Highlight mathematical language such as ‘rotation’, ‘reflection’, etc.)>>
<<Total number of possible figures using 4 cubes is 8. Further information about this task, as well as picture of the possible solutions, can be found on page 12 of the Paying Attention to Spatial Reasoning document>>

21. Spatial Reasoning Skills - Revisited
Navigating & wayfinding
Imagining objects move in space (mental rotation and transformation)
Composing / decomposing
Partitioning
Visualizing
Non-verbal reasoning
Scaling up or down
Manipulating objects
Creating or designing objects
Moving one’s body in space
Disembedding
Comparing
Perspective taking
Shifting dimension
Diagramming
Locating objects/ remembering locations of objects
Orienting
Let’s go back to this list of skills and think about which skills you used in the cube activity. <<Give participants time to do this individually. Discuss any highlights or aha comments.>>
Creating & reading maps, graphs, visual data
Unitizing (same space & shape)
Proportional Reasoning

22. Let’s Build! Each person selects 7 cubes.
Build a three dimension irregular figure so that all cubes are connected.
Then, on grid paper, draw the:
Front view
Top view
Side view
Allow participants to build a figure using seven cubes and draw the perspective images using the grid paper.
<<Have them share their perspectives with an elbow partner to confirm accuracy and to discuss any challenges. Discuss whole group any comments people wish to share.>>
<<image taken by module author; no copyright issues>>

23. Top, Front & Side Views Set 1
Visualize
Verbalize
Verify
Top, Front & Side Views Set 1
SIDE VIEW
FRONT VIEW
TOP VIEW
Now, using the cubes you have, build the figure that would have these three views. <<Allow time to complete this.>> Compare your structure with your table mates.
With acknowledgement to Moss, Bruce, Flynn & Hawes

24. Top, Front & Side Views Set 2
Visualize
Verbalize
Verify
Top, Front & Side Views Set 2
FRONT VIEW
SIDE VIEW
TOP VIEW
Now, using the cubes you have, build the figure that would have these three views. <<Allow time to complete this.>> Compare your structure with your table mates.
With acknowledgement to Moss, Bruce, Flynn & Hawes

25. Top, Front & Side Views Set 3
Visualize
Verbalize
Verify
Top, Front & Side Views Set 3
TOP VIEW
SIDE VIEW
FRONT VIEW
Now, using the cubes you have, build the figure that would have these three views. <<Allow time to complete this.>> Compare your structure with your table mates.
With acknowledgement to Moss, Bruce, Flynn & Hawes

26. Spatial Reasoning Skills - Again
Navigating & wayfinding
Imagining objects move in space (mental rotation and transformation)
Composing / decomposing
Partitioning
Visualizing
Non-verbal reasoning
Scaling up or down
Manipulating objects
Creating or designing objects
Moving one’s body in space
Disembedding
Comparing
Perspective taking
Shifting dimension
Diagramming
Locating objects/ remembering locations of objects
Orienting
Let’s go back to this list of skills. At your table, use the sticky notes to record where you see spatial reasoning in all of the mathematics strands in the curriculum for your grade. There is one copy of the curriculum document on the table if you wish to use it, but it is not necessary. Provide one example per sticky. <<Give participants time to do this individually.>> At your table, pile up your sticky notes to identify common elements across grades/strands.
<<Discuss some of the connections cross strand and across grades.>>
Creating & reading maps, graphs, visual data
Unitizing (same space & shape)
Proportional Reasoning

27. Resources
edugains.ca
On the Paying Attention to Mathematics Education page, there is an Adobe Presenter which allows you to learn more about Spatial Reasoning. Thank you for you participation today.