To each his own. All roads lead to Mecca. There’s more than one way to skin a cat. We’ve got plenty of ways to describe people being able to find their own path to the same place, and the path of least resistance to a solution is something massively valued in mathematics. Elegance and efficiency are worthwhile aims, but not all methods support progress to higher levels of maths.
My new role as a school-based tutor means I do lot more close reading of student maths as it happens, and on the whole I’ve been trying to let students use their own methods to cut down confusion. No more ‘whack it in a box’ for anything to do with proportionality and multiplicative reasoning unless the student’s got nowhere to start.
I used to think it doesn’t matter what they do if it ‘makes sense to them’. Teach them 3 ways to do division and let them pick the method that they like, understand or feel most comfortable with.
I’m not convinced anymore. We should be teaching the methods that allow students to access as much of the curriculum as possible.
All teachers have pedagogic opinions on what’s more effective in maths. I love a grid for most things and I find chunking heinous. I think discussing these things with people who don’t agree is one of the most interesting and intellectually stimulating aspects of being a maths teacher. Others disagree. There’s a lot to argue about for a subject that’s often said to be unique in virtue of its definite answers.
The language used varies from teacher to teacher. We could ‘balance’. We could ‘get x on its own’. We could ask ‘what’s 1 x?’.
Some people describe processes I’m convinced come from a random word generator, like ‘float and ping’, ‘crossing the river’ or ‘gouging the onion’. (That last one is a lie)
One of my tutees solves equations like this:
They start with the unknown, then follow BIDMAS (another thing for another day) until they get to the ‘answer’ of 18. Then they work backwards with inverse operations to find the value of the unknown.
I thought this was brilliant. It was accessible. It linked with something they already knew with the BIDMAS and the student felt confident with it. It feels a bit like dividing fractions with ‘keep, flip, change’, but I’m there to get them through an exam (again, another thing for another day).
Then it dawned on me that this wasn’t going to work for much beyond the most simple one-step equations. This method is not going to support progress in linear equations, never mind beyond.
Aside from the cacophony of threes, the negative coefficient can become problematic, as does what to do next in my tutee’s method.
This BIDMASsy arrows thing could still work if you rewrite the equation from 3=30-3x to 3=-3x+30.
But 30-3x being equivalent to -3x+30 is a big leap for lots of kids. I seem to spend my life reminding kids subtraction isn’t commutative, and now I’m saying that thing with a negative symbol can be written either way round?
The only way I approach this is to think of collecting terms, as below, but then this depends on students being able to navigate the links between negatives and subtractions, which for weaker students adds another dimension of difficulty to solving equations instead of fitting in with what they already know.
The other way to rewrite the equation would allow the student to crack on with their arrows too:
None of this is wrong per se, as it would enable the student to solve the equation, but learning BIDMASsy arrows for one thing, and then “adding/subtracting unknowns to both sides” is using two different methods for one question. It feels clunky and all over the place. “If it’s negative you have to add to both sides before your start” is only a whisker away from “gouge the onion” in terms of meaning if it doesn’t fit into established understanding.
Students learning methods for every case is ineffective and makes maths more difficult in the long run, as it downplays any flexibility of thought and we end up with hundreds of baseless ‘rules’ with no context.
When I teach solving equations I have post-it notes on a metre stick and do lots of flailing if we don’t apply the same operation and magnitude to each side.
I’m calling it ‘Slapstick for Learning’, and it’s coming to an INSET near you soon.
I want them to be focusing on the truth of the equality instead of what the unknown is. When we first introduce solving equations students can often solve them by inspection.
“Yeah but it’s 3 though”
I don’t think this is being able to solve an equation though. It isn’t enough. They can solve that equation.
By focusing on the equality,”Is this still true?” becomes the question rather than “what do I do?” or “What’s x?”
This way, students can practise some substitution, and then when we get to solving inequalities and we need to know if our sign needs changing it’s not another inaccessible idea. It’s all linked.
By manipulating unknowns on both sides at this stage students will be more prepared for unknowns on both sides, or even to solve minefields like:
In this case, the negative coefficient of x, the negative solution and sometimes even the fact the 19 is on the left side of the equals sign can cause students to stop dead in their tracks and we haven’t even got to equations with unknowns on both sides of the equality yet.
If we know we can do things to that equation that keeps the equality intact, students can now focus on what option would be useful. By framing it this way they can start to have a go at manipulating the equation to see where they get to, and when they might need to go in a different direction, but this too relies on prior knowledge, by being comfortable with algebraic notation and the effects of operations on expressions.
In the first equation, 3x+6=18, I can discuss with students whether or not they would factorise first and divide, when would they add the constant, when wouldn’t they, give me an example of an equation where it would make sense to factorise first, give me an example where it wouldn’t, etc. By approaching teaching equations in this way, students can begin to reflect on what they could do when they get an equation, which is the first step towards what they will do. This is problem solving.
Both fluency in manipulating algebra and the standards unit box diagram allow you to tackle problems across a wide-range of content in algebra and proportionality.
When we talk about students being flexible in their approach to solving equations, I take that to mean their algebraic manipulation. They can preserve the equality of the equation to get to where they need to be and to discern the most efficient way, even at the level of simple two-step equations like 3x+6=18.
That principle of equality becomes essential for solving harder equations and if students are secure with it, solving inequalities and changing the subject becomes an extension of their previous learning rather than ‘another topic’.
In the long run, that’s better for students’ problem solving skills, self-efficacy and resilience. Learning 3 superficial methods and picking one isn’t.
Questions that increase in difficulty one niggle at a time are great for this, and Dave Taylor‘s website, increasingly difficult questions, is as much a planning tool for honing explanations and concept-building as it is a bank of quality exercises.
I’m sure there’s lots of things I teach in a way that allows students to answer questions and solve problems at the level of challenge they’re at then, but might cause problems later on, particularly at AS/A2 level.
Maths teachers of year 11s up and down the country are getting tactical on what they do between now and exams. When is it alright for a quick and dirty method? I want to say never, but realistically I think sometimes there’s a trade-off for time.
Of course, two days later I taught a paired session and I completely changed my mind on this when it came to expanding brackets, but that too is another thing for another day.