Illustrations are better when ....

I was wondering how far we should develop learners understanding of the distinction between the two types of addition, namely aggregation and augmentation.

But additionally, (excuse the pun) I’m also interested in how well some of the diagrams we use to illustrate addition, support the concept or detract from understanding.

particular, the active element of the addition operation.

Let’s take (oops another pun) 2 + 3 as an example.

How successful are these in conveying the meaning to which we attribute 2 + 3?

Fig 1

Fig 2

Fig 3

Fig 4

Fig 5

Fig 6

Is any diagram better than another?

Do any of these create possibly unintended misconceptions and do any lend themselves more to augmentation rather than aggregation or vice versa?

Let’s consider aggregation, which rightly or wrongly I see as the opposite to partitioning, or in more child-like parlance “breaking into parts” and “putting together”.

Debbie Morgan & the NCETM make ample use of this diagram.

Fig 7

Am I wrong in feeling this is best used actively with manipulatives such as counters being moved from circle to circle to demonstrate the actual putting together or breaking up a sum, but less comfortable when the illustration shows both before and after? So that in fact twice as many counters are on show.

Fig 8

The diagram above attempts to illustrate the movement of counters to show the before and the after in the aggregation of the two parts to make the sum, rather than using an illustration such as below that in shows both states in one diagram.

Fig 9

And yet I have less qualms about using the diagram with the numbers (rather than counters) in each circle.

Fig 10

Is this a legitimate concern?

Similarly using “multi-link” cubes or similar can be used in a practical context to illustrate the aggregation and partitioning of a value and so two distinct states of equal value that can be show abstractly as

5 = 2 + 3

So why do I have reservations?

Perhaps it’s because the diagram does not show all the practical manipulation of the concrete objects that needs to take place to develop the child’s understanding.

So, a diagram like this,

Fig 11

.. can successfully support learning if it is coupled with this active manipulation by learners and not just presented on its own.

Similarly, in the case of augmentation, where perhaps there is a more distinct “before” and “after” the illustration can be enhanced by an arrow to suggest something has happened.

Fig 12

An operation has taken place. In this case, three counters have been added that weren’t there before (unlike the case of aggregation).

But actively using manipulatives would make this clearer rather than just a passive and inactive diagram.

I suppose the question remains, should we make the distinction between aggregation and augmentation?

But if aggregation (addition as joining together) is the opposite of partitioning, then augmentation (adding to) is the opposite of “taking away”. Which may then help to strengthen the concept of at least one element of subtraction.

I’d certainly welcome comments before embarking on number lines. !!