On Horizontalness and Verticalness

Over the last couple of days, I have been having a very interesting discussion with James Dixon, a numeracy learning specialist from Australia, about the use of vertical versus horizontal representations. It stemmed from James’ post on LinkedIn which included an article (that is definitely worth reading if you work with primary age mathematics learners, and that some secondary teachers may also find useful) about things to consider before introducing “vertical addition”; which we might call in the UK the “formal addition algorithm”.

It was question 2 that got me thinking about the role of horizontalness and verticalness in early addition and subtraction calculations – “Should addition number sentences be represented horizontally before vertically?”. The main thrust of the article here is that the rigidity brought in by teaching vertical addition limits pupils’ use of, and receptiveness to, alternative strategies (as well as some interesting research in point 3 about single-digit versus multi-digit conception). This made me wonder, “Would this still happen if all approaches to addition and subtraction were modelled vertically?” So instead of writing:

3 + 5 = 8

We would write:

   3

+ 5

    8

Even when first introducing learners to the concept of addition (and similarly for subtraction).

Of course, I don’t just mean writing vertically in the abstract, I also mean modelling vertically with our concrete and pictorial representations, so instead of this:

We would show this:

Instead of this:

We show this:

Instead of this:

We show this:

(I considered having the number line increase going downwards there, but I thought that would be a step too far for most people – although see my note on variation later on).

If we did this, right from the very beginning of pupils learning about addition and subtraction, would this stop some of the issues that James highlights in his article?

Now, of course, I don’t know the answer to this having (a) never worked with children that young in an educational setting, and (b) not being aware of any school or other educational setting that has been this intentional about the orientation of the representations they use rather than just their choice of representations themselves. However, I can see some potential benefits as well as at least one potential drawback (although I may not be seeing all of the drawbacks given my unfamiliarity with the age range where this is first taught).

The obvious benefit is that the writing of all sums then mirrors what happens when we start to use the addition/subtraction algorithm, meaning that (at least potentially) this transition is less marked than it perhaps currently is when pupils move from mental strategies for addition and subtraction to written calculations involving algorithms. If, for example, I have learnt that I can sum 9 and 5 using a strategy such as:

Then this at least could mean that when I come to move onto the algorithm, I am already familiar with writing sums this way and so it is only left to understand why I start with the ones column and progress up the powers of 10. Even if I am working with larger numbers but using a “decomposition” strategy rather than the algorithm, such as carrying out 342 + 251, I can model these vertically rather than horizontally:

Which has the potential added benefit that this mirrors how a number like 251 would be decomposed on a Gattegno or place value chart.

This, however, is also the source of the potential drawback that I can see – would setting out the symbolic record for every addition and subtraction strategy vertically rather than limiting verticalness to the algorithm risk confusing these approaches. My heart wants to say “no” provided that these strategies are fully secured and embedded using appropriate concrete and pictorial representations, but my head tells me that things are rarely so simple.

There is also the nagging bit of me that reminds me that I am writing the sums vertically from top to bottom, but using the number line vertically from bottom to top. I have a strong feeling that this disconnect might cause issues, although I also think that we should be varying the orientation of the number line regularly, as well as the direction of increasing value, lest pupils form the misconception that this is an essential rather than non-essential property of the number line. Variation theory tells us that deliberately varying the essential and non-essential features of a concept is part of what leads to a deeper understanding of the concept and so I can see how, at an appropriate point, it would actually be beneficial for learners to see and work with number lines that run both horizontally and vertically, and where the direction of increasing value is up, down, left and right. This could potentially be done after the number line is introduced, completely separately to its use in helping to model addition and subtraction strategies, so that by the time pupils see a number line paired with an addition or subtraction approach they are comfortable working with number lines in all configurations and so having a number line that increases in value in a downwards direction would not be a particular barrier to using it to make sense of the strategy.

Another possible benefit to this way of writing calculations would be the use of the equal sign. It is well documented that pupils often harbour misconceptions about the number line being an operator or instruction meaning “provide the result of”, which is due in part to them answering endless questions of the form “3 + 4 = 7”. Whilst recognition of this, and possible remedies for it (such as writing calculations leading with the equal sign, e.g. 7 = 3 + 4”) are becoming more prevalent, the above vertical representations have the benefit of using the equal sign in a way that shows the calculations are equal rather than simply the result. In fact, the use of the equal sign is actively avoided for the statement of the result of the calculation. This should then help to reinforce the meaning we want pupils to take about the equal sign, namely that it tells use about the relationship between different mathematical expressions.

Finally, it would potentially solve some issues when it comes to working with algebraic expressions. As a secondary teacher, I know that there is often confusion that arises when simplifying expressions such as 3x + 5 – 2x + 4, with pupils often subtracting incorrect terms in the expression. Of course, a good manipulative or representation helps enormously with this sort of thing, however if the first thing a pupil thought to do when faced with something like this was to re-write it vertically, like:

    3x

+ 5

  – 2x

+ 4

This might help pupils with identifying more clearly which terms are added and which terms are subtracted from the total.

Like I say, there might be drawbacks I haven’t spotted here, but absent someone pointing them out to me I think there is enough in this for teachers to at least consider changing their way of symbolic recording of calculation from a default horizontal to a default vertical, and would be very interested to hear about the experiences that people have if they do (or already have).