“Sir… before, I never used to understand area, but now I do.”
Year 10 boy
Simplicity is king. The above phrase was uttered after I found a way to reduce something deceptively complex, into something very simple. At the end of my last post I suggested that the complexity of what we wish to teach, in any subject, must compel us to seek out the simplest possible explanation. Not long ago I reduced the sprawling and insidious complexity of area to a single thought; here’s how.
In October we taught a unit on area. I felt I spent the first week doing it very badly, and even by the end of the unit, wasn’t overly happy about it. Looking back over some of the end of unit results for my Year 8 top set I realised it went better than I thought, with a solid level 10a (if I haven’t mentioned it before, I have the rare and remarkable pleasure of teaching an actual genius) three level 9s, six 8s and five 7s. So I was wondering why in my head it was such a disaster of a unit, until I recalled the 10 remaining results that were below my floor target of 7c, including four 4s! That’s over a third of the class let down by me, not nearly good enough.
Area… it’s a funny one. It should be so easy, just follow a certain series of steps, and out pops the area for a shape; the processes are perfectly straight forward. All I’m looking to do here is teach kids how to calculate the area of shapes, the processes, the algorithms. It’s what’s known to the inner-circle of maths teachers as ‘instrumental understanding.’ There are those who claim that it’s easier for pupils to remember how to find the area of shapes if you first show them where the processes or algorithms come from – don’t just show them ‘how’, show them ‘why’. There’s a stack of reasons why I think that’s not a good idea, but within the context of this post it would be sufficient to say that that’s certainly not simple! Valuable to learn at some point, certainly, perhaps even essential, but it’s not Holy Grail of recall and application (see ‘Why is it that students always seem to understand, but then never remember?’)
So area should be straightforward, why isn’t it? In short, there are five processes to remember for the five basic 2D shapes, and that’s before we even get into compound shapes, surface area, curved surface areas, sectors and segments, and also before we start looking at the many ways in which those five shapes can be twisted and turned in their presentation. Without the ‘relational understanding’ of where the processes come from, it’s actually very difficult for people to compartmentalise which process goes with which shape – hence the argument for trying to teach that understanding. The complexity in this topic is not in the application of the process, that part is exceedingly simple, but instead in holding lots of different processes in the mind simultaneously, and trying to recall which one goes with which shape! But then, since trying to teach the relational understanding directly is – I’m going to be bold enough to hazard – doomed to failure, what’s the alternative?
The first four basic 2D shapes are the rectangle, the parallelogram, the triangle, and the trapezium. There are three parts to this tale:
Part 1 – Shaky Start
I model how to find the area of each shape. Most kids replicate without issue, though for others their fear of maths paralyses them. We’ve learnt a few shapes now over several lessons, we move on, we come back to these early shapes and practise questions with them mixed together, many can’t recall what to do when.
Part 2 – No Child Left Behind
Determined not to let them down, after we moved onto the next unit I institute ‘literacy starters,’ in which a child is chosen at random to answer, as a full sentence, a question such as ‘How do we find the area of a trapezium?’ I told them exactly what to say in response; if we want to use fairly loaded language we could say they ‘parroted’ back the responses. This is what I asked them to say:
- “To find the area of a rectangle, multiply by the length by the width.”
- “To find the area of a parallelogram, multiply the length by the perpendicular height.”
- “To find the area of a triangle, multiply the base by the height, and then divide by two.”
- “To find the area of a trapezium, add the parallel sides, multiply by the perpendicular height and then divide by two.”
This actually worked wonders. Doing this in nearly every lesson over months – also looping in questions from other units as we progressed – had the group to a point where everyone could respond effortlessly; practice makes perfect. But was it meaningless words they said, were they simply parroting? This is what I found: when they would need to find the area of a triangle for example, they might still not remember what to do and put their hand up to ask me. When I came over, I simply asked them ‘how do you find the area of a triangle?’ they would be about to respond automatically, before realising they knew exactly how to solve their problem, and so continued with that instead.
When teaching a later unit on volume, I realised how to dramatically simplify the topic, and I also used it to heavily practise finding areas. By now, everyone was much more proficient than at the time of the area unit.
Part 3 – D’oh!
Many months after October I was asked to teach a lesson to a Year 10 intervention group. This group had around fifty students in it preparing for an exam, with maybe 6-8 teachers to support them. I was asked to teach them area. I was concerned. Many of the kids in this group were extremely weak, and even many of those who actually weren’t were still asking if you add sides together when attempting area questions, or spasmodically trying to divide or multiply by two when finding the area of a rectangle; such confusion, and here I was asked to teach them everything they needed in a single lesson, while several colleagues observed!
How can we clear the fog of complexity?
I thought about how I’d simplified volume, and how successful that had been, and realised that it was possible to do the same thing for area, and that so much of that modelling, and those long complicated sentences my Year 8s spent so long repeating, were like trying to jump up over a high wall when they’d been a convenient ladder nearby all along! D’oh!
So here it is, the area of all four shapes expressed as a single thought:
“Area is always, always, always, nothing more than length times height.”
In explaining this I would usually trace out a horizontal line with my hand when saying ‘length’ and a vertical line when saying ‘height,’ to add a visual cue. In this way, all of area is reduced to just two simple lines. Everything else is ignored. If shapes are presented rotated from their default appearance, just ask kids to rotate their paper til it looks like the parallelogram they know and love, simple!
Does this embed misconceptions? I don’t think so; area is always length times height – actually, in finding any length there must always be the multiplication of two perpendicular dimensions, but you try saying that to a novice and see the response! (I did try, didn’t go down well.)
So with the Year 10s I introduced the rectangle and parallelogram, and explained the above. They then practised.
Then I introduced the triangle and trapezium and noted that:
“For the two shapes that begin with a T, the triangle and trapezium, you have to divide by two.”
There’s lots of repetition of this line, and emphasis on the ‘t’ sound, and lots of waving of two fingers. Does this embed misconceptions? I’m honestly not sure – might mess with someone if they ever take a look at the surface area of a trapezoid, but really these are aide-memoires, crutches to be used and removed as appropriate, training wheels to be discarded as early as possible, not rigorous and logically constructed proofs!
One last note, as seen in the diagram, the length of a trapezium is weird. It has a weird name, and a weird length. You have to add on the top bit as well to get the ‘real’ length, but that’s all; much simpler than A = ½(a + b)h.
Length times height. Length times height. Length times height. Every time, all they have to remember now, is length times height. It gets rid of all the other jargon words – no more worrying about widths, bases, perpendiculars, slants and whatnots, just length times height, every time. Oh there may be the odd extra you have to do afterwards, but always start with length times height. The 72 words of my original, technically correct phrases, reduced to just three words, length times height.
After the Y10 lesson, on his way out one young man stopped to say “Sir… before, I never used to understand area, but now I do.” Does he? Well no; what he does understand though, or at least know, is how to calculate area. He didn’t know that before, and best of all it felt easy to him, it was simple. That’s a huge win, and it’s one that can be built upon.
This was one lesson in preparation for an impending exam, but I wonder had I done that with my Year 8 group at the start, perhaps I could have covered every shape in one lesson, and perhaps every child would have understood what I asked them to do. From there, we would have had more time free to start thinking about some of the ‘whys,’ and maybe even reintroduce some of the technical vocabulary. ‘Length times height’ can even be extended to cover circles:
Now it gets really interesting, because what started as a reductionist explanation is actually becoming something more; it’s starting to expose pupils to the concept that, in the parlance, circle area is still a function of the multiplication of two perpendicular dimensions, rather than the arbitrary squaring of the radius. In the beginning though, just three words: simplicity is king.