Solving Linear Equations: some thoughts

This is quite a difficult one to examine for a methods of last resort blog post, as many methods exist for solving linear equations and different teachers will use different approaches. Probably the two most common are some form of balancing approach, similar to this:

and then the use of function machines, similar to this:

Now inherently there is nothing wrong with either of these approaches (except the function machine solves 3a - 5 = 19, not the given equation 3a - 5 = 10) provided pupils understand why they are carrying out the operations they are, or how the function machine relates to the equation they are solving, and then how the inverse machine relates to the original equation. 

The slight problem I have with both of these is how open they are to a more procedural approach. I can imagine a lot of teachers falling into the trap of teaching how to find the inverse function as a procedure rather than with any real understanding. I can imagine lots of teachers showing pupils how to manipulate both sides of an equation whilst keeping them in balance, but without imparting any real sense of why what they're doing works or what the purpose of the whole affair is. Equally I can imagine this not being the case and these methods both being taught well. 

Recently I have begun to consider a different approach, although I haven't really used it extensively yet. In the main I have drawn attention to it when using balancing as a way of developing understanding, or when pupils have suggested incorrect statements when solving an equation. The approach I have used looks at a sort of 'If...then' or 'what follows?' kind of approach. I will illustrate below with and example:

Solve the equation 3a - 5 = 10:

If 3a - 5 = 10, what follows?

Well if 3a - 5 = 10 then it follows that 3a = 15.

If 3a = 15, what follows?

Well if 3a = 15 then it follows that a = 5.

I am very aware of a couple of big points when it comes to this:

              1) What if a pupil gives something that does follow but isn't useful, for example, if 3a - 5 = 

                   10 then 3a - 15 = 0?

              2) What if pupils do not have the understanding of relationships and operations necessary to

                    see what follows, for example if 3a = 14 then a = 4⅔.

.

In response to the first I would (and frequently do, even when using balancing) allow this to go. I would then explore the consequences of this and try and eventually show how that wasn't a useful step. Over time I would want to develop an understanding in pupils of what the next useful thing to write would be, but in the beginning I wouldn't necessarily be too worried about this. I wonder if allowing pupils to explore (under very controlled conditions obviously) the consequences of making true but not useful statements would actually help them develop their understanding of the concept of equality and equation solving. It would concern me if we always limited pupils to the correct next step in the reasoning, as this would seem to then smack of becoming a procedure we expect pupils to follow. In fact I would strongly consider having an entire lesson early in secondary school where rather than solve equations, pupils simply have to write other true equations based on the original. I have seen activities and sessions like that being used already and I can definitely see the merit in them.

In response to the second, I would simply say that this is worthwhile diagnostic information, as it points to a gap in pupil understanding of a more basic concept. If this was the case it is a clear indication to me that I need to go back and do more work on fractions and inverse operations as the pupil in question clearly doesn't have the requisite procedural fluency in these areas. Hopefully with the advent of mastery teaching situations like this would become rarer as times goes on.

Ultimately no matter what approach you are going to use for solving linear equations, I would urge you to be wary of falling into the trap of explaining the 'how' without ever getting near the 'why'. There are ample opportunities when using balancing to explore why one statement leads to the next, and in function machines why the different machines link to each other, as well as to the original equation. However for me equations are about the relationship between two equal quantities, and I wonder if the focus on operations that is part of both balancing and function machines obscures this somewhat, so I will be exploring the use of the approach I have outlined - and I would welcome feedback from others who may be using or thinking of using similar ideas.