Spatial Reasoning

I was at a Ministry-run conference last week working with other Math-folk from all over Ontario and suddenly had an epiphany regarding Spatial Reasoning. To try and understand what Spatial Reasoning is, have a look at this video and predict which of the five squares below the paper will look like when it is unfolded:

I haven't come across a precise definition of what Spatial Reasoning is but in general, most folk agree that it involves visualising, perspective taking, mental transformations, composing and decomposing (shapes, numbers, measurements, data, and algebraic expressions). Nora S. Newcombe in this excellent article says 'Spatial thinking concerns the locations of objects, their shapes, their relations to each other, and the paths they take as they move.' She goes on to show how Spatial Reasoning is not a 'learning style'  but a habit of thinking, that for anyone it can be improved, that whilst there may be gender differences, the important fact is not the causation of these differences but that both genders can still improve their Spatial Reasoning.

My big lightbulb moment was when I realised that Spatial Reasoning is so much more than geometry. Indeed it transcends Math. It is a way of thinking that helps us solve problems in number, measurement, algebra, geometry and data handling; in science (figuring how the atoms in a molecule are arranged); in technology (any work on perspective or visualising how to build a certain structure); in sports (reading the shape of an opponent's defence or making a 'no-look' pass); in the arts (seeing the sculpture in a block of marble or the visualising of dance moves that a choreographer must make; in geography (any map making or map reading activities); in history (visualising what a building must look like from the clues found in an archaeological dig).
This got me wondering, though, how aware are we as teachers of Spatial Reasoning and much time do we spend on it? Working with some colleagues at my table we quickly came up with the following ideas just in Maths:

Primary:

• Use positional language as much as possible
• I'm thinking of a shape that looks like a triangle and a rectangle stuck together. Draw what it might be.
• There is a shape in this bag. Feel it but don't look at it. Now tell me what you think it is.
• Imagine holding a can of soup. How many circles might you see?
• I have a letter. When I turn it upside down, it still looks the same, What might it be? What couldn't it be?
• You measured the table width with your pencil and it was 10 wide. My pencil is twice as big as yours. In your head, imagine me measuring the width. Would I use more, less or the same as your pencil?

Junior:

• I have two rectangles and put them together. What shape could I end up with?
• I cut of the corner of a square. Use a geoboard to show the shape I cut off and the shape that I'm left with.
• I have joined 6 cubes together. From the front, they look like an I. From the side, they look like a T. What do they look like from above?
• Visualising the mean as an evening out of all the scores as shown in this post.
• Visualise two congruent triangles. How could you arrange them to make a parallelogram? How does this help us get a formula for the area of a triangle?

Intermediate:

• Imagine two congruent trapezoids. How could you arrange them to make a parallelogram? How does this help us get a formula for the area of a trapezoid?
• Imagine unfolding a cylinder. What shapes will you see? How would you work out the area of these shapes.
• Imagine two lines. One crosses at (-4, 0) and (0,4). The other crosses at (0,-2)  and (2,0). How would these lines look?
• Imagine completing the square like this

Senior:

• I have a function that has four roots and a range y<4. What could my function be?
• Imagine you slice a cone. What are the different shapes that the cross-section could be?
• Imagine you have three different planes. How many different ways could they intersect?
• How many zeroes are at the end of 125!
• What happens to the secant through two points in a curve as the first point gets closer to the second?

In fact, I will go so far to say that visualising techniques (far more so than rote memory of rules and formulae) are essential in understanding calculus.

After the session, I engaged in a wonderful Twitter dialogue with Malke Rosenfeld who is also learning about the importance of Spatial Reasoning. I highly recommend her blog here.

And as for the solution to the paper folding exercise? Check below!