This has to do with breaking apart, reconstructing (while conserving) and repackaging. In maths, it is said to be the foundation for number sense and understanding number relationships.
Part-part-whole relationships involve seeing numbers as being made up of two or more parts. It has been shown to increase understanding of place value, number concepts and generally help students with word problems.
I love Singapore maths (my latest fancy and research area) where children start with concrete objects, then move on to pictorial representations, then on to abstract.
There are a lot of online articles and references to PPW relationships but this is how I teach it based on my students’ developmental progression (i.e. class or age).
My Preparatory class (2-3yrs): strictly a perceptual stage
1. Before we start learning to identify numbers from 1 to a maximum limit of 5, we start PPW relationships using 2-3 parts flat or 3D jigsaw puzzles. It could be a 3D coloured shape or a wooden animal jigsaw (we would have learnt about the animal prior to this class). This would teach them to know that parts can be combined to form whole or a whole can be broken down into parts.
2. When we do start identifying numbers, to maintain the relationship of part to whole concept, I prefer to use fingers for my pupils. First, we identify the representation of finger to its number, and I work on one for a while because any finger up alone can be a one, not just the fore-finger. Fingers on one hand first (remember our limit is five) until we get to baby-complex stage – e.g. when 2 fingers are up, 3 are down. It always works for me and they tend to catch on quickly (maybe it’s because we all have five fingers together on one hand). I leave our objects for the older pupils.
Nursery one class (3-4½yrs): predominantly perceptual but we move to figurative.
1. We are old enough to work with 2 hands here, so we move to a limit of 10. From simple to complex,
a. We work with fingers first e.g. right hand up (five fingers) and two fingers on the left is the “same as” seven fingers on a wide hand (join the two hands together).
b. We start using 5 parts to 12 parts jigsaws (based on child’s ability) for fun.
c. We move to counters and part-whole sheets like the one below for numbers between 0-10 based on level of each child.
d. Then we move to 2 single 5 frames
e. Then single 10 frames.
Nursery 2 class (4-5½yrs): mainly figurative stage
We work extensively with single and double ten frames. Combining, decomposing and breaking up number while conserving the topic number. We still use the part-whole sheets extensively and branching to sheets with 4 parts, instead of three. Be careful not to treat it as addition and subtraction yet. Using an abacus (ten in a row with two colours like the one below) will also help reinforcement. So too will dominoes.
By the time the child moves to grade one, he would be able to work with number bonds (questions involving number bonds can be found in our Quantitative Aptitude textbooks) for further understanding, and later on fractions and word problems in the middle/upper primary grades.