What Does the '=' Sign Mean?

Here is a video of two students in a Grade 9 Academic class talking through their solution to

              -15+x= -x+15.

As you listen to it, can you figure out the misconception?

Despite the otherwise articulate description ("... you need to isolate the variable..." etc.) there is a real fundamental (yet common) algebraic misconception here which is:

when you bring a term to the other side, it switches sign.

I've been wondering when such misconceptions begin. More to the point, I'm wondering if students truly appreciate the meaning of the '=' sign. If we reduce algebra to a bunch of tricks that needs to be memorised ("... when the number jumps over the gate it trips and so it changes sign...") then these might cause these misconceptions.
Some research that Christine Suurtamm presented at the OMCA conference in February also shines light on a common misconception. Students were asked to solve the following problem:

8+4=?+5

This is a breakdown of how different grade responded:

Notice how in each group the most common response is 12. Is this because students translate the '=' sign as 'Now do what you have just read' (in this case 8+4)? If the students have only experienced questions of the type 8+4=? and not, say 8+?=12, then it is no surprise that they make this mistake.
Notice also how that none of  the Grade 5s and 6s (in this study) got the actual answer correct. Is this because their Math experience is now more about following rules and less about understanding the concept? Have these students really developed the concept that the '=' means that the expressions either side of the sign are balanced? It is crucial that they do this. In the book Mathematical Misconceptions by Anne Cockburn and Graham Littler, the point is made that equality is a central but a neglected concept and that there is a strong correlation between the ability to understand the '=' sign and the ability to solve equations.

Even seeing how numbers can be decomposed and recomposed will help with this. For example, asking students to think of how many different ways 12 animals can be split between 2 fields would generate a lot of answers of the type that we can write as 12=8+4 (as opposed to 8+4=12).
Or if I put 5 green marbles and 9 black marbles in one side of a set of scales and 3 red in the other side, students will see that the pans are not balanced. I can ask how many blue marbles do I need to add to make the scales balanced? There are a variety of ways to solve this and when I consolidate the students' thoughts I can tell them that they have really solved this equation

5+9=3+?

Or written another way:

                                        14=3+x

With younger grades, they will rely on their number sense to solve this not some contrived (and partially understood) rule. Activities like this will help students really understand the meaning of '='.
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There are some great thoughts and ideas on equality in the Ministry of Ontario's Paying Attention to Algebraic Reasoning document, especially on pages 6 and 7.

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!