What does simplicity look like?

“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.

maeda_slide_1

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?

18

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:

  1. “To find the area of a rectangle, multiply by the length by the width.”
  2. “To find the area of a parallelogram, multiply the length by the perpendicular height.”
  3. “To find the area of a triangle, multiply the base by the height, and then divide by two.”
  4. “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!

fog

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!

Rectangle Parallelogram

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!

Triangle Trapezium

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:

Circle 2

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.

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About Kris Boulton

Teach First 2011 maths teacher, focussed on curriculum design.
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15 Responses to What does simplicity look like?

  1. David Thomas says:

    This is really interesting. What’s remarkable is that the simplicity of the statement “Area is just length times height” actually leads to far greater relational understanding of what area is than does the list of formulae. It’s all about perpendicular measurements that create square units, something that’s often an incidental side effect in area lessons.

    I’m not 100% sure yet about how I’d sequence this to make clear the differences between the shapes. My inclination is to teach rectangles, then parallelograms, then trapezia and finish with triangles. What did you do?

    I think this is a really fascinating approach to teaching concepts, and puts a much-needed emphasis on the language used to explain an idea. I wonder how successfully you can boil down other concepts to a short phrase, and still retain the underlying mathematics at its core.

  2. krisboulton says:

    ““Area is just length times height” actually leads to far greater relational understanding”

    Did you catch @dazmck ‘s tweet to me, suggesting that students might better be able to ‘guess’ at the formula for the area of an ellipse when seeing it this way? Never even occurred to me – brilliant addition. Start with simplicity, and carefully build the layers of relational understanding.

    If you showed them the 4-squares over the circle next, they’d probably then be able to surmised that a similar demonstration with an ellipse would involve 4 rectangles, from which you could begin to draw out the relationship between squares and circles, in that they are special, regular cases of rectangles and ellipses.

    ***
    Since I’ve only taught this once, and just in a single lesson to Y10s with an exam two weeks away, I showed them the rectangle and parallelogram together, and then they practised. Next I showed them the triangle and trapezium together, then they practised. Finally I showed them the circle. In this case I didn’t use ‘length times area’ for the circle – they needed the ‘radius times radius’ vocabulary, since diagrams weren’t going to have two lines on them in the exam.

    Compound areas I decided against teaching – those who were able would either figure it out, or get it with a little 1-1 support from one of the many teachers. Those who weren’t, were not going to be helped by an extended explanation from me at the board!

    @redorgreenpen made a really astute observation, that a triangle is a trapezium where a = 0. I suppose in a proper lesson you could teach the trapezium first, then use an IWB to show the top side getting smaller until it’s a triangle, and so draw out that relationship. …I feel like it might disrupt the simplicity of ‘length times height’ though… I think I’d maybe want to show them that after they’d practised and embedded their ability to calculate the areas.

    ***

    As for other topics:

    In the volume unit I looked all the questions they needed to answer, and realised that, assuming they knew how to calculate areas, and compound areas, about 80% of them could be boiled down to ‘Volume is area times length’

    I kicked off with a little Engelmann-inspired sequence of examples and non-examples to communicate what prisms were. A few Y8s were able to then articulate a definition for a prism quite clearly. In immediately assessing their understanding, I discovered a mistake in my sequence, since a majority of the pupils opted for saying that a sphere was a prism – my non-examples hadn’t properly communicated the need for consistency in x-section, but it was great to see so cleanly that that was 100% my mistake, and not theirs, as Engelmann would like us to accept.

    My Year 7 group is very weak compared with 8.1. They still understood what a prism was (could state whether something was, or wasn’t a prism) and a couple felt like they could articulate a definition; they gave it a really good try, but didn’t quite get there.

    I didn’t talk about ‘cross-sectional area’ until I felt it was the right time – so I left it as just ‘area’, let the diagram do the talking, then drew out the important distinction between cross-sectional area and surface area later.

    When it came to pyramids, I noted how they were a bit smaller than a prism of the equivalent height, so the volume would have to be less, so we’ll divide it by 3, and you *always* divide by 3. I didn’t, but suppose I could have said that you divide by 3 because it’s in 3D, rather than dividing by 2 for certain 2D shapes. Is that embedding a misconception, or is it just making it easier to remember what to do, when? In fact with volume, I seem to remember a discussion in which it might very well be because it’s a 3D shape that one needed to divide by 3 – something related to integration I think. It’s the same for a sphere, you need to divide by three at some point in finding its volume.

    In a recent unit on Pythagoras and Perimeter I tried to boil most of the unit down to one of two things:

    Diameter times Pi
    or
    Pythagoras’ Theorem (we spent a lot of time on this alone, looking at how to articulate it in different ways – other than the formula, I can’t think of anything simpler than practice)

    Seems to have worked out mostly for geometry units thus far, but then I think geometry naturally lends itself to ‘problem solving,’ and problem solving is taking a few things that you do know, and combining them to derive new things that you don’t. By comparison, algebra has so many rules and special cases to learn, I haven’t yet found a better way of teaching it than incremental complexity and shed loads of practice.

    I think I did try to boil it all down to three things eventually:

    1) = means ‘whatever is on the left of me, is equal to, whatever is on the right of me
    2) A solution to an equation ‘protects’ the equals sign (one Y8 lad helpfully noted that we were like body guards then, seeking to protect the equals sign by finding a valid solution)
    3) To either side of an equation you can add, subtract, multiply or divide by anything you like, provided you do the same to the other side

    With algebra I wonder if the ‘complexity’ lies more in undoing all the damage done by primary education! Trying to convince kids that the equals sign doesn’t just mean ‘give the answer’ took a long time, then there’s the constant problem that they want to evaluate something and write ‘the answer’ to the right of it, rather than underneath. From one line to the next you can do one of three things – evaluate numbers and operators on either side, manipulate the expressions on either side, or rearrange the equations across the equals sign. This is so utterly counter-intuitive to everything they’ve done for 7-8 years prior that it’s no wonder some of their faces suggest it’s a painful experience! Would we have fewer problems with algebra if primary school questions were set up more like this:

    Evaluate:

    3 x 5
    A: 15

    Rather than:

    Work out:

    3 x 5 =

    I’m going off on a tangent… and shall stop.

    • David Thomas says:

      On area: I’d never thought of the triangle/trapezium relationship before. I agree though, that’s it’s more an interesting observation to look at later than a way of introducing the area of a triangle. I think introductory sequences have to be simple and only focus on the key elements – everyone needs to understand them. Something like that, although interesting, risks leaving people behind and them not being able to start work.

      On algebra: I got our Year 8 to do a week on the equals sign this year. They just did tasks where equals didn’t mean “write the answer here”, and where they had to balance both sides by making different sums. It was time really well spent, as all teachers were gobsmacked by how little understanding students actually had of equality.

      Simplifying algebra is hard because lots of things are conventions. However I do think we could do better at teaching equations as balancing right from the start. Function machines have always bothered me, as they are so limited in scope and don’t get students into the right habits. Manipulating algebra is generally about asking “what do I have to do to isolate this variable?”, and so that’s the question we need to be teaching from the start. This would make algebra more of a linear path, and remove the Level 5-6 jump.

  3. @dazmck says:

    Great conversation! It’s refreshing to find blog posts about specific content rather than more general issues of pedagogy, all the more so in maths.

  4. liamwilkinson says:

    Agree with @dazmck great posts.

  5. Max says:

    Fascinating approach – I can see the appeal. I’m uncomfortable with the idea that area is ALWAYS length x height – does it transfer adequately to new contexts? (e.g. area of a football field – I suppose it relies on a student being able to manipulate the football field into the orientation they recognise in their head).

    I think you already know the answer as to whether: “For the two shapes that begin with a T, the triangle and trapezium, you have to divide by two” embeds misconceptions – you talk about that in the post. Is this just a crutch? That depends on a student getting to the point where they do actually understand the concept of area, and so can leave this crutch behind. My fear is that it would be a rule not a crutch to them because they’d never get to the point where they could leave the crutch behind (and know enough to do so).

    I admire the drive for simplicity though – perhaps the best thing about this approach is that if you really can cover all four shapes in one lesson, you then have time to develop the ‘whys’, as you say.

  6. krisboulton says:

    The biggest sticking point for some of the maths teachers I most admire, seemed to be the worry that this is suggested *in place of* developing a relational understanding. I’ve had an almost identical conversation with different people of the last few days in other subject domains – such as whether or not to teach French while using terms such as ‘the imperfect tense,’ or just call it the ‘WING tense’ like Michel Thomas does, since in English it’s the tense where you say ‘I Was doING’ or ‘We Were doING’ etc.

    The concern people have I think is one of content, and a fear that the understanding will never come; that I’m advocating we replace it with a return to the dark old days of raw instrumental understanding. Not at all! My current argument is one of correct sequencing. I strongly suspect, for a number of reasons, that ‘teaching relational understanding’ is not the panacea that many hoped. The arguments are actually very complex, and certainly nuanced, and to some extent I’m still formulating them and figuring out where the boundaries lie myself. But certainly, I’m not suggesting that algorithms should be advocated INSTEAD OF relational understanding.

    I think that length x height is actually pretty fool proof; there will always need to be a multiplication of two perpendicular dimensions in calculating an area. In the end it just depends what you define as length and height! There’s no real reason that we have to talk about length and width when talking about a rectangle, it’s pretty arbitrary. Other terms that have value, such as ‘perpendicular height’ or ‘base length’ can certainly be reintegrated at a later point, and so they should be! Once again it’s a question of sequencing, not of content.

    Amongst these complexities I mention, there’s one probably worth drawing out. I did talk about them as a crutch ‘to be discarded’ in the post, as it makes things easier for people to accept often. In reality, there is again something more nuanced at work here. The example of dividing fractions is perhaps a good one. I know one primary school to be having remarkable success with the Singapore Maths curriculum which has their Y5s dividing fractions, something that I’m told is usually left until Y7. They do it using diagrammatic representations to arrive at the correct result. This is great, and it helps one to really grasp what the division of fractions means, however it has a very real flaw – it’s crazy slow. Compared with the KFC algorithm (keep the first fraction, flip the second, change the sign) it’s very, very slow. Not only that, but KFC is actually essential for rapid manipulation of algebraic fractions. But, if only ever taught KFC, one might not have the same picture in their mind as to what division of fractions really is. My point here is that, actually rather than discarding one or the other, sometimes the ideal situation is to eventually hold *both* in mind, and use whichever is appropriate for a particular situation. A simplification like ‘length x height’ could actually take someone a long, long way. Later understanding of course that really we’re talking about two arbitrarily labelled perpendicular dimensions will be ideal, however length x height can still be a useful, since it’s simpler and therefore faster to think about, provided one understands its limits. Most mathematicians actually thinking in terms of ‘flip and ping’ when rearranging equations – it’s extremely fast! But it has seriously drawbacks, and of course an expert knows where the limits lie, and switches between appropriate strategies at the right time.

    Maybe a better example is the T thing – it’s a silly mnemonic to help them remember in the early days; but then actually, it helps one sort out their thoughts so quickly that you might never want to abandon it! Sure, you want to reach a point where you understand that it’s mere *coincidence* that they begin with a T and involve a division by two, *absolutely*! But having understood that, you might still well use the idea to recall what to do when, because it still remains a handy mnemonic.

    So that paragraph turned into a little more than I hoped. This is why I initially intended to leave it as ‘complex and subtle things,’ to be discussed another time perhaps… but anyway, it’s written now, there it is, digest at will!

  7. Max says:

    The point about sequencing makes a lot of sense, and helps to put the whole post in perspective. Is it fair to paraphrase what you’re saying as: the goal is relational understanding, but that instrumental understanding might be the best way to get there?

  8. krisboulton says:

    Almost – I define it as a ‘*necessary* first step.’ More precisely, I interpret Willingham’s article to be saying that.

    “4. Remember that inflexible knowledge is a natural step on the way to the deeper knowledge that we want our students to have:”

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  12. Victor Minkov says:

    I wonder what you think of this: A colleague of mine teaches area from the trapezium rule and extrapolates its use to the other shapes. Square/rectangle/parallelogram/wrongbus (they notice that a = b, so (a+b)/2 = b (bxh). Trapezium as normal. Triangle, a=o as you mentioned above yielding bxh/2

    From what I can see, that method reduces the memory load down to one formula. I’m not sure it extrapolates well to the circle. Maybe if one considers a and b to be horizontal radii and h as vertical, so that (a+b)2 = r, h = r, hence r-squared.

    Now that the trapezium rule is no longer included in the exam formula, starting with it could be one way to embed it via rehearsal.

    • Kris Boulton says:

      Sounds like it has merit. It would depend on what you’re shooting for to some extent though… as a means of expressing the relationships between shapes it sounds solid. As a means of recalling the different formulae I agree, I think it could be useful, but I’d want to see it working in practice – does it work, or do pupils find it too complicated as as introduction? Too much mental effort to get from that starting point to calculating the area of a parallelogram, say.

      Then there’s the final point – in reality these formulae are all so simple that they should be recalled automatically, without any need for derivation – I feel as though straight forward varied drill practice distributed over time would pull that off (retrieval effect.) So while I think it’s always good to have an extra means of verifying whether something’s been recalled correctly, and it’s certainly important for the relationships in mathematics to be recognised, I wouldn’t want recalling how to calculate areas of polygons to always involve an act of derivation.

      I suspect it would fail with *most* pupils in primary years, or even Year 7 and 8, making this something better to bring in later to express the relationships between the shapes and their areas, once the standalone formulae are known.

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