## Wednesday, March 19, 2014

### 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.