It's well established that good mathematics teaching helps students to connect concrete, visual (or pictorial) models and abstract models.
What are the fundamental visual models for the four operations in calculation?
To help understand the answer to this question I think it's first necessary to say what they are not. Consider this excellent poster by Maria Droujkova. It provides a lovely insight into the kind of ways in which multiplication appears in nature. But it does not explain how people 'see' multiplication in their heads. There are clearly two separate steps. In the first they decode that a situation requires multiplication by recognising a multiplicative structure. In the second they carry out the multiplication. They may complete this second step using known facts or an algorithm they cannot explain. But if they are confident and have been well taught they should be able to draw a picture which explains their thinking. And except in very simple cases for multiplication this is most likely to be a picture/diagram of a repeated addition process. Repeated addition is the most widely understood and used mental model for addition.
I think that the visual models used for the four key operations with real numbers are as follows. In every case students benefit from being able to express these models with real objects, a number line or equivalent thinking and with base 10 materials.
Addition
Count all
Subtraction
Take away
Difference
Inverse of addition
Multiplication
Repeated addition (the use of array/multiplication through areas of rectangles is powerful for scaffolding variety of thinking and future work in alebra)
Scaling
Division
Splitting/How many each(for 1)
Chunking
Inverse of multiplication
Reciprocals
Notes on models (in italics) which are generally not taught and/or clearly understood.
I've shown two models in italics because they are currently weakly defined in Western teaching however I've listed them for completeness because students may be using them and if we do not try to accept and acknowledge what they are already doing complications often arise. Also there's no harm in being aware of what others do.
The scaling model for multiplication requires that if any general quantity (such as a length or a weight) is considered to be 1, another number (such as 7) can be estimated. Cuisenaire rods have been used to explicitly develop students' skills in scaling. It seems to be very well developed as a strategy in non-literate cultures (see for example Nuhnes' work with street traders). It also seems to bear some resemblance to the teaching of music through doh ray me rather than absolute note names. To work properly each interval needs to be drilled and many results are learned.
The model of reciprocals for division (and multiplication) uses the awareness that multiplying by a number is the same as dividing by its reciprocal and vice versa. e.g. multiplying by 1/3 is the same as dividing by 3. Dividing by 2/3 is the same as multiplying by 3/2 (or 1.5) and so on. This is demonstrated in LiPing Ma's comparative study of mathematics teaching in China and the US as being an active strategy used in China.
These are the models which emerged in my classroom when I used the SDPQ activities decribed in my 8-part blog - How do the Chinese do it? They also fit with my extensive reading of the current literature in mathematics education and my discussions with other enthusiasts. But if your perception differs please do say. None of this is set in stone. Different people may have different models and there may be things I've missed.
