Monday, 15 July 2013

Disambiguating the Equal Sign ("=")

Disambiguating the Equal Sign ("=")

As a somewhat competent mathematician* I have been trained to view the equal sign as a firm barrier between the LS (left side) and RS (right side) of an algebraic statement and more importantly as a balance point between these two sides. Namely, whatever is on one side had better "equal" whatever is on the other side of we are fraught with mathematical (and logical) contradictions.

For example, the statement: 
3x + 5 = 7x + 3

 makes mathematical "sense" in this context if and only if x has the value 1/2. 

This idea of the equal sign is useful in this kind of context, where we (or more likely, our students) are being expected to solve an algebraic system; in this case a fairly simple one. It gives rise to "rules" like whatever you do to one side you have to do the other, and so on and so forth. The idea of these being to avoid the contradictions that would result in giving you a value for x that doesn't work the original statement. To get from one to the other mathematical principles are combined with logical reasoning of the "if something is true in situation A, it must also be true in situation B; because situation B can be shown to have equivilent requirements on the variables as situation A" sort. 

As I noted above, it is not us, the mathematicians who are doing this in my head, it is our students. Meaning I am first and foremost a teacher and that mathematical context may not be shared with them, the students. Here is where the ambiguous idea of the equal sign comes in because students have different contexts, and the one context that is very important for students understanding of math that is thought of differently for mathematicians and old school mathematically trained teachers** is that of the calculator. 

For many students who I have taught or tutored = is actually synonymous to the "Enter" key on a computer. It means "OK Go! Onto the next step now." After all, on a calculator you often push the = button between each mathematical step you are doing. For students in this context,

2 + 7 = 9 * 5 = 45 - 3 = 42 / 7 = 6

makes mathematical sense. It is even possible to follow the steps that students followed from first to last (first was to add 2 and 7, last was to divide by 7). At no point along this path does = declare that the LS and the RS are the same. Here = is shorthand for breaking up each step in the process. 

As math educators I think it is important to work to understand what our students are doing and why. To look for the things they do get and not jump to declare something as "unclear" or "not correct." I'm not saying that this use of = is always applicable or that is superior to the traditional mathematical rigor context. In fact, I think it's important to notice that the equal sign as a barrier is important for algebra, for the manipulating of equations and solving statements; but in the context of a string of simple calculations it may actually be easier and more clear to simply write out what you are doing. It would save time and perhaps allow students who have trouble "showing their work" to put down what they did and in the order they did it. I can also see it being a non-trivial assignment to write a full algebriac statement for the second example. 

The math conventions are there for a reason, most of them good reason; but there are situations where we have been trained to only see things when they are laid out in the fashion we are used to and to see gibberish otherwise. We need to try to be conscious of when it is useful to teach our students those conventions, when it's not, and when it's more important to see the work through their eyes.

How many times have you seen this type of thing on student work? What else have you previously graded as "unclear" or "incorrect" but have learned to see differently?

*I did my undergraduate degree in mathematics and opted for a specialist in "pure" math; which put a focus on the algebra, calculus, and analysis areas of math.

**I'm relatively young, but I count myself among this old guard because I (mostly) got math and found it easy to see the calculator as one tool for the doing of maths. In the context of this post, I also really connect with the idea of = as a hard and fast "every must be equal on both sides" idea and have corrected my students on this in the past. I have however come to see this differently when in the teacher role of trying to assess students work, students thought process, and grading how they got to an answer as being a correct process or not.